text
stringlengths 701
483k
| target
stringlengths 239
1.88k
| evaluation_predictions
sequence |
|---|---|---|
the primordial cosmic microwave background ( cmb ) was generated when photons first decoupled from the baryonic fluid when the universe was only 400,000 years old .
the vast majority of these photons travel unperturbed to the present day , and features of their angular power spectrum such as acoustic peaks and the damping tail @xcite record valuable information about cosmological parameters @xcite .
baryons and dark matter evolve from small inhomogeneities at decoupling into increasingly complicated large - scale structure which can subtly perturb the observed pattern of cmb anisotropies . assuming that the primordial cmb is gaussian , non - gaussian correlations in the observed map can be used to reconstruct the intervening large - scale structure @xcite .
in addition to the importance of learning about the large - scale structure itself , reconstruction of the weak - lensing potential generated by structure is essential to constraining tensor perturbations .
weak lensing converts a fraction of the e - mode polarization generated by scalar perturbations at the last - scattering surface into b - mode polarization in the observed map . only by subtracting this b - mode polarization
can one conclusively detect the primordial b - modes which serve as a model - independent signal of tensor perturbations @xcite .
understanding lensing reconstruction requires a more detailed discussion of how weak lensing affects the cmb .
weak gravitational lensing deflects the paths of cmb photons as they travel from the last - scattering surface to the observer .
this deflection is accomplished by a projected lensing potential which is a weighted line - of - sight integral of the gravitational potential between the observer and the surface of last - scattering . at each point on the sky , lensing remaps the temperature and polarization to that of a nearby point at the last - scattering surface , the deflection angle being the gradient of the aforementioned projected lensing potential . assuming that this deflection angle is small
, the temperature at any point can be expanded in a taylor series in the gradient of the lensing potential . in fourier space
, this expansion appears as a series of convolutions of individual temperature and projected potential modes .
the observed temperature - squared map in fourier space also appears as a convolution of individual fourier modes .
subject to an overall normalization dependent on the scale of the fourier mode , these convolutions cancel in such a manner that each fourier mode of the temperature - squared map acts as an estimator for the _ same _ fourier mode of the projected lensing potential .
lensing reconstruction as outlined above has been considered previously @xcite . in these works ,
two sources of noise were identified , and a filter of the temperature - squared map in fourier space was chosen to minimize the variance associated with lensing reconstruction subject to these noise sources .
the first source is intrinsic signal variance ; the observed large - scale structure is one arbitrary member of an ensemble of realizations allowed by theory .
the second source of noise , endemic to this method of lensing reconstruction , is a consequence of the nature of the primordial cmb . like the large - scale structure itself ,
the pattern of cmb anisotropies at the last - scattering surface is only one of many possible realizations allowed by theory .
we do not know _ a priori _ which of these realizations nature has provided us , and this uncertainty hinders our ability to deconvolve the effects of lensing from true anisotropies at the last - scattering surface .
even if the true pattern of anisotropies at the last - scattering surface was known , the finite amount of power in the cmb at small scales would still constrain lensing reconstruction .
silk damping at the last - scattering surface suppresses cmb power at small scales , while the finite resolution of any real experiment would limit the detection of any signal that is present at small scales .
lensing reconstruction fails below scales at which there is sufficient power , for the same reason that any remapping is indistinguishable given a uniform background . here
, we consider a third source of noise neglected in previous studies .
the filtered temperature - squared map is an unbiased estimator for the lensing potential in the approximation that a correlation between two given temperature modes is induced only by the single lensing mode whose wavevector is the sum of that of the two temperature modes .
in actuality , any combination of two or more lensing modes whose wavevectors sum to this total induce correlations between the two temperature modes .
there are many such combinations , but since these correlations add incoherently we do not expect a systematic bias . nonetheless , for estimators of each individual lensing mode we must use our knowledge of other lensing modes to subtract off this unwanted bias .
this is an iterative process , and since our knowledge of the lensing map is imperfect it induces noise in lensing reconstruction .
we calculate this additional variance for various estimators constructed from cmb temperature and polarization maps , and show how it compares to the dominant noise sources for an all - sky cmb experiment with a noise - equivalent temperature of 1 @xmath1k @xmath2 . since the lensing - potential power spectrum is a measure of the theoretical uncertainty with which we can predict the value of a given lensing mode , this noise associated with lensing reconstruction causes a systematic overestimation of the lensing - potential power spectrum .
this systematic bias must be accounted for in order to compare observations with theoretical predictions .
this paper is organized as follows . in
[ s : lensing ] , we define the formalism we will use to explore the effects of weak lensing on the cmb .
the taylor expansion of the lensed cmb map in gradients of the lensing potential is given in both real and fourier space , and the power spectra and trispectra of various components of the cmb temperature map are listed for later use . in
[ s : estimators ] , we show that the fourier modes of the temperature - squared map when properly filtered can serve as estimators for the fourier modes of the displacement map with the same wave vector . using the power spectrum and trispectrum given in the preceding section , we calculate the variance associated with this estimator including a new component neglected in previous studies .
this variance is evaluated numerically using the currently favored @xmath0cdm cosmological model with baryon density @xmath3 , matter density @xmath4 , cosmological constant density @xmath5 , hubble parameter @xmath6=0.65 , and power - spectrum amplitude @xmath7 .
we then use the displacement estimator for individual fourier modes to construct an unbiased estimator for the lensing - potential power spectrum in [ s : power ] , and calculate the variance and covariance associated with this estimator .
a few concluding remarks about the implications of our work for future studies are given in
[ s : disc ] .
the appendix contains useful formulae related to additional estimators of lensing based on polarization and a combination of temperature and polarization .
we consider weak lensing under the flat - sky approximation following refs .
@xcite . as discussed before @xcite
, weak lensing deflects the path of cmb photons resulting in a remapping of the observed temperature pattern on the sky , @xmath8 \nonumber\\ & \approx & { \theta}({\hat{\bf n } } ) + \nabla_i { \phi}({\hat{\bf n } } ) \nabla^i { \theta}({\hat{\bf n } } ) + \frac{1}{2 } \nabla_i { \phi}({\hat{\bf n } } ) \nabla_j { \phi}({\hat{\bf n } } ) \nabla^{i}\nabla^{j } { \theta}({\hat{\bf n } } ) + \ldots\end{aligned}\ ] ] where @xmath9 is the unlensed primary component of the cmb in a direction @xmath10 at the last scattering surface .
the observed , gravitationally - lensed temperature map @xmath11 in direction @xmath10 is that of the unlensed map in direction @xmath12 where @xmath13 represents the lensing deflection angle or displacement map .
although a real cmb map will include secondary contributions such as the sz effect @xcite , we assume that such effects can be distinguished by their frequency dependence @xcite .
they will not be further considered in this paper . a noise component denoted by @xmath14 due to finite experimental sensitivity
must be included as well .
thus the total observed cmb anisotropy will be @xmath15 .
taking the fourier transform of the lensed map @xmath11 under the flat - sky approximation , we write @xmath16 where @xmath17 + \frac{1}{2 } { \int \frac{d^2 { \bf l''}}{(2\pi)^2}}{\phi}({{\mathbf{l^{\prime\prime } } } } ) \\ & & \quad \times { \phi}({{\mathbf{l}}}- { { \mathbf{l^{\prime}}}}- { { \mathbf{l^{\prime\prime } } } } ) \ , ( { { \mathbf{l^{\prime\prime}}}}\cdot { { \mathbf{l^{\prime } } } } ) \left [ ( { { \mathbf{l^{\prime\prime}}}}+ { { \mathbf{l^{\prime}}}}- { { \mathbf{l}}})\cdot { { \mathbf{l^{\prime}}}}\right ] + \ldots \ , . \nonumber\end{aligned}\ ] ] cmb correlations in fourier space can be described in terms of a power spectrum and trispectrum as defined in the usual manner , @xmath18 where the angle brackets denote ensemble averages over possible realizations of the primordial cmb , large - scale structure between the observer and the surface of last - scattering , and instrumental noise .
the subscript @xmath19 denotes the connected part of the four - point function and the superscript @xmath20 denotes the temperature map being considered ( @xmath21 or @xmath22 ) .
the lensing - potential power spectrum can be defined analogously , @xmath23 where here the angle brackets denote an average over all realizations of the large - scale structure .
we make the assumption that primordial fluctuations at the last - scattering surface are gaussian .
gaussian statistics are fully described by a power spectrum ; the gaussian four - point correlator , @xmath24 is zero .
the instrumental noise @xmath22 is also assumed to be gaussian , as is the lensing potential @xmath25 .
this second assumption is justified because the dominant contributions to the lensing potential come from intermediate redshifts @xmath26 at which linear theory holds .
using these definitions , we can calculate the anticipated power spectrum and trispectrum of the observed cmb map .
because the instrumental noise is uncorrelated with the signal , the power spectrum of the observed map is the sum of signal and noise power spectra , @xmath27 the power spectrum of the noise component is given by : @xmath28 where @xmath29 is the fraction of the sky surveyed , @xmath30 is the variance per unit area on the sky , and @xmath31 is the effective beamwidth of the instrument expressed in terms of its full - width half - maximum resolution @xmath32 . a cmb experiment that spends a time @xmath33 examining each of @xmath34 pixels with detectors of sensitivity @xmath35 will have a variance per unit area @xmath36 @xcite .
the power spectrum of the lensed cmb can be determined by inserting eq .
( [ e : thetal ] ) into eq .
( [ e : specdef ] ) as discussed in @xcite , @xmath37 \ , c_l^{{\theta}{\theta } } + { \int \frac{d^2 { \bf l}_1}{(2\pi)^2 } } c_{| { { \bf l}}- { { \bf l}}_1|}^{{\theta}{\theta } } c^{\phi\phi}_{l_1 } [ ( { { \bf l}}- { { \bf l}}_1)\cdot { { \bf l}}_1]^2 \ , .
\label{e : lenpow}\end{aligned}\ ] ] this result is given to linear order in the lensing - potential power spectrum @xmath38 .
lensing neither creates nor destroys power in the cmb , but merely shifts the scales on which it occurs as seen by the fact that @xmath39 the observed cmb trispectrum can be calculated in a similar manner ; under our assumptions of gaussian instrumental noise and no secondary anisotropies the trispectrum of the lensed component @xmath40 is the sole contribution to the total observed trispectrum , @xmath41 [ ( { { { \bf l}_1}}+ { { { \bf l}_3 } } ) \cdot { { { \bf l}_4 } } ] + c^{{\phi\phi}}_{|{{\bf l}}_2+{{\bf l}}_3| } [ ( { { { \bf l}_2}}+{{{\bf l}_3 } } ) \cdot { { { \bf l}_3 } } ] [ ( { { { \bf l}_2}}+{{{\bf l}_3 } } ) \cdot { { { \bf l}_4 } } ] \big ] + \ , { \rm perm . } \ , .
\nonumber \\
\label{e : trilens}\end{aligned}\ ] ] the term shown above is manifestly symmetric under the interchange @xmath42 , while the `` + perm . ''
represents five additional terms identical in form but with the replacement of @xmath43 and @xmath44 with the other five combinations of pairs .
the total trispectrum is symmetric under the interchange of any given pair as one would expect .
having established a formalism within which to analyze weak lensing , we now consider the problem of reconstructing the lensing potential from an observed cmb temperature map .
in this section , we examine lensing reconstruction following the approach of ref .
@xcite , largely adopting their notation as well .
the only important difference in notation is that we use @xmath40 to denote the lensed temperature field and @xmath45 for the unlensed field following ref .
@xcite and most recent papers . ref .
@xcite uses the opposite convention . for @xmath46 and to linear order in @xmath25 , @xmath47 where @xmath48 and @xmath49 .
note that @xmath50 differs from the unmarked @xmath51 that first appeared in eq .
( [ e : specdef ] ) in that it denotes an ensemble average only over different gaussian realizations of the primordial cmb and instrument noise ; a fixed realization of the large - scale structure is assumed .
for the purposes of estimating the large - scale structure actually realized in our observable universe , this is the appropriate average to take to ensure that our estimators are truly unbiased for a typical realization of the primordial cmb . when calculating the noise associated with lensing - potential estimators and again for power spectrum estimation in [ s : power ] , we will return to the full unmarked ensemble average .
( [ e : ea ] ) , an immediate consquence of eq .
( [ e : thetal ] ) , suggests that a temperature - squared map appropriately filtered in fourier space can serve as an estimator for the deflection field @xmath52 .
hu and okamoto define five different estimators for the deflection field constructed from various combinations of the temperature and polarization ; we discuss the temperature - squared estimator in this section and relegate the analogous formulae for polarization estimators to the appendix .
the minimum - variance temperature - squared estimator derived in ref .
@xcite is @xmath53 where @xmath54 @xmath55^{-1 } \
, , \ ] ] and @xmath56 .
substitution of eqs .
( [ e : ea ] ) , ( [ e : bias ] ) , ( [ e : filt ] ) , and ( [ e : norm ] ) into eq .
( [ e : est ] ) shows the desired result , @xmath57 namely that @xmath58 is indeed an unbiased estimator for the deflection field in fourier space .
we now proceed to calculate the variance of this estimator . at first
, we assume a complete knowledge of all lensing modes not examined by this estimator . in that case
, we find @xmath59 where @xmath60 . evaluating the four - point function in the integrand of eq .
( [ e : estvar , cmb ] ) to second order in the lensing field , we obtain @xmath61 \nonumber \\ & & \quad \times \bigl [ { { \mathbf{l^{\prime}}}}\cdot ( { { { \bf l}_2}}- { { \mathbf{l^{\prime } } } } ) \bigr ] - \frac{1}{2 } { \int \frac{d^2 { \bf l'}}{(2\pi)^2}}{\phi}({{\mathbf{l^{\prime } } } } ) { \phi}(-{{\mathbf{l}}}- { { \mathbf{l^{\prime } } } } ) \bigl\ { c_{l_1}^{{\theta}{\theta } } ( { { { \bf l}_1}}\cdot { { \mathbf{l^{\prime } } } } ) \bigl [ { { { \bf l}_1}}\cdot ( { { \mathbf{l}}}+ { { \mathbf{l^{\prime } } } } ) \bigr ] + c_{l_2}^{{\theta}{\theta } } ( { { { \bf l}_2}}\cdot { { \mathbf{l^{\prime } } } } ) \bigl [ { { { \bf l}_2}}\cdot ( { { \mathbf{l}}}+ { { \mathbf{l^{\prime } } } } ) \bigr ] \bigr\ } \biggr ] \nonumber \\ & & \quad \times \biggl [ \bigl ( c_{l_{1}^{\prime}}^{{\theta}{\theta } } + c_{l_{1}^{\prime}}^{{\theta}{\theta}{{\rm n } } } \bigr ) ( 2 \pi)^2 \delta_{{\rm d}}({{\mathbf{l^{\prime } } } } ) + { \phi}({{\mathbf{l^{\prime } } } } ) { f_{\theta\theta}}({{{\bf l}_1 } } ' , { { { \bf l}_2 } } ' ) - { \int \frac{d^2 { \bf l'}}{(2\pi)^2}}c_{l^{\prime}}^{{\theta}{\theta } } { \phi}({{{\bf l}_1 } } ' - { { \mathbf{l^{\prime } } } } ) { \phi}({{{\bf l}_2 } } ' + { { \mathbf{l^{\prime } } } } ) \bigl [ { { \mathbf{l^{\prime}}}}\cdot ( { { { \bf l}_1 } } ' - { { \mathbf{l^{\prime } } } } ) \bigr ] \nonumber \\ & & \quad \times \bigl [ { { \mathbf{l^{\prime}}}}\cdot ( { { { \bf l}_2 } } ' + { { \mathbf{l^{\prime } } } } ) \bigr ] + \frac{1}{2 } { \int \frac{d^2 { \bf l'}}{(2\pi)^2}}{\phi}({{\mathbf{l^{\prime } } } } ) { \phi}({{\mathbf{l^{\prime}}}}- { { \mathbf{l^{\prime } } } } ) \bigl\ { c_{l_{1}^{\prime}}^{{\theta}{\theta } } ( { { { \bf l}_1 } } ' \cdot { { \mathbf{l^{\prime } } } } ) \bigl [ { { { \bf l}_1 } } ' \cdot ( { { \mathbf{l^{\prime}}}}- { { \mathbf{l^{\prime } } } } ) \bigr ] + c_{l_{2}^{\prime}}^{{\theta}{\theta } } ( { { { \bf l}_2 } } ' \cdot { { \mathbf{l^{\prime } } } } ) \bigl [ { { { \bf l}_2 } } ' \cdot ( { { \mathbf{l^{\prime}}}}- { { \mathbf{l^{\prime } } } } ) \bigr ] \bigr\ } \biggr ] { \rm + perm . } \nonumber \\\end{aligned}\ ] ] the terms given explicitly in eq .
( [ e:4pt , cmb ] ) correspond to the correlations between @xmath62 and @xmath63 and those between @xmath64 and @xmath65 .
the `` + perm . '' stands for two additional terms , identical in form , arising from the pairings @xmath66 and @xmath67 .
this expression indicates how uncertainty in the cmb at the last - scattering surface propagates into uncertainty in lensing reconstruction for a particular realization @xmath68 of the large - scale structure .
to linear order in eq .
( [ e:4pt , cmb ] ) , correlations between the modes @xmath69 and @xmath64 are induced by those lensing modes whose wavevectors are the sums of any pair of wavevectors of these four modes .
these lensing modes are precisely those forming the diagonals of the quadrilaterals depicted in fig .
[ f:2quad ] . in practice
, we do not know the large - scale structure between us and the last - scattering surface , so we assume a variance given by eq .
( [ e : lenpowdef ] ) with a model - dependent power spectrum @xmath70 .
we must average eq .
( [ e : estvar , cmb ] ) over different realizations of the large - scale structure ( denoted by @xmath71 ) to obtain the total expected variance of our estimator , @xmath72 where @xmath73 the assumption that the lensing potential is gaussian imposes the constraint @xmath74 which closes the quadrilaterals of fig
. [ f:2quad ] . the average of the four - point correlation function in eq .
( [ e : estvar ] ) can be calculated by further averaging eq .
( [ e:4pt , cmb ] ) over the large - scale structure .
terms linear in the lensing field vanish when averaged over different realizations of the large - scale structure .
quadratic terms in the lensing field arise as products either of two linear terms or of a zeroth and second - order term .
averages over the product of two linear terms produce the connected part of the four - point correlation function , the trispectrum defined in eq .
( [ e : specdef ] ) .
averages over the product of a zeroth and second - order term have no connected portion , but instead furnish an implicit dependence on @xmath70 in the total observed power spectrum of eq .
( [ e : temptot ] )
. the final result of averaging over the large - scale structure can be expressed in terms of the observed power spectrum and trispectrum , @xmath75 \ , , \nonumber \\\end{aligned}\ ] ] where the trispectrum can written in terms of @xmath76 as @xmath77 this form of the trispectrum is consistent with that of eq .
( [ e : trilens ] ) given directly in terms of power spectra . since @xmath78 , @xmath79 and the first term of eq .
( [ e:4pt ] ) vanishes .
the remaining two terms containing pairs of delta functions , inserted into eq .
( [ e : estvar ] ) , yield the dominant contribution to the variance , @xmath80 \ , , \ ] ] where @xmath81 .
notice that @xmath82 is zeroth order in the lensing potential @xmath25 ; it depends on the lensing potential power spectrum @xmath83 only implicitly though the total observed power spectrum @xmath84 .
the ellipsis represents terms of higher order in @xmath83 that we now proceed to calculate .
these terms arise from the trispectrum term of eq .
( [ e:4pt ] ) after @xmath85 is removed . substituting these results into eq .
( [ e : estvar ] ) , we find that to first order in @xmath83 , @xmath86 \ , , \ ] ] where @xmath87 is given by , @xmath88 the first - order contribution to the noise @xmath87 involves integrals over the lensing - potential power spectrum , and thus probes lensing modes with wave vectors different from that of the estimator @xmath58 .
it can be interpreted physically as interference from these other modes in the determination of the mode @xmath89 being estimated .
the filter @xmath90 was chosen to optimize the signal - to - noise ratio in the absence of the first - order contribution @xmath87 ; it is no longer an optimal filter once this additional noise is taken into account . as long as @xmath91 ,
the noise reduction that can be attained by re - optimizing our filter will not be significant .
formulas analagous to those presented here relevant to the construction of estimators using polarization data are given in the appendix .
the significance of @xmath87 for two different experiments is shown in fig .
[ f : plrec ] using the currently favored @xmath0cdm cosmological model with baryon density @xmath3 , matter density @xmath4 , cosmological constant density @xmath5 , hubble parameter @xmath6=0.65 , and power - spectrum amplitude @xmath7 .
the planck experiment is equivalent to a one - year , full - sky survey with temperature and polarization sensitivities of 12.42 and 26.02 @xmath92 respectively and resolution @xmath93 arcminutes as described in [ s : lensing ] .
the reference experiment has the same resolution but superior sensitivities of 0.46 and 0.65 @xmath94 for temperature and polarization .
these estimates of experimental parameters are identical to those given for the planck and reference experiments of ref .
the @xmath95 and @xmath96 estimators have noise power spectra intermediate to those of the @xmath97 and @xmath98 estimators , while the @xmath99 estimator has substantially higher noise because the primordial cmb lacks true b - modes in the absence of inflationary gravitational waves .
we see that for planck , with its comparatively inferior polarization sensitivity , the @xmath97 estimator will be best although it will be unable to detect individual fourier modes of the deflection field at the @xmath100 level .
the reference experiment , and further experiments with similar sensitivity and even higher resolution , should be able to push @xmath100 detection of individual @xmath89 modes to @xmath101 by primarily relying on the @xmath98 estimator . in these cases
the secondary noise @xmath102 is only smaller than the dominant noise @xmath103 by a factor of a few , whereas for higher sensitivity ( noisier ) experiments like planck it is smaller by at least an order - of - magnitude .
this illustrates an interesting point , apparent from fig .
[ f : plrec ] , that the zeroth - order noise @xmath82 declines dramatically with decreasing sensitivity until it becomes dominated by cosmic variance while the @xmath87 is largely unaffected by instrument sensitivity .
the reasons for this trend are that instrument noise appears in @xmath81 through its contribution to the denominator of @xmath104 as shown by eqs .
( [ e : filt ] ) and ( [ e : norm ] ) .
decreasing instrument noise raises the value of @xmath104 thereby lowering @xmath82 .
by contrast instrument noise is reflected in @xmath87 through its effects on both @xmath105 and @xmath104 as shown in eq .
( [ e:1st ] ) .
smaller instrument noise raises @xmath104 as before , driving @xmath87 up in this case , but this is compensated by a decrease in @xmath105 which appears as a prefactor outside the integrals .
these two effects largely cancel each other out , rendering @xmath87 remarkably insensitive to instrument noise .
although complete reconstruction of the deflection field @xmath89 can be an enormously powerful tool , such as for b - mode subtraction @xcite , for some purposes estimates of the lensing - potential power spectrum @xmath106 are sufficient .
this power spectrum is a model - dependent prediction of theories of large - scale structure formation , and therefore estimates of the power spectrum from real data could be used to test these theories as well as the consistency of other determinations of cosmological parameters .
furthermore , since estimates of all the modes @xmath89 with @xmath107 can be combined to estimate @xmath106 , @xmath100 detection of the power spectrum can be pushed to much higher @xmath108 than can that of individual modes .
the deflection - field estimator @xmath58 derived in the preceding section can be used to construct an estimator for @xmath106 .
our first guess for an appropriate lensing - potential power spectrum estimator is @xmath109 where @xmath110 is the area of the sky surveyed and @xmath111 is an annulus of radius @xmath108 and width @xmath112 .
we ensemble average our estimator over different realizations of the cmb and large - scale structure using eq .
( [ e : prodave ] ) by bringing @xmath113 from the left to the right - hand side .
this yields @xmath114 \ , .\ ] ] the definition of the dirac delta function , @xmath115 implies that @xmath116 . furthermore , in the limit that @xmath112 is small compared to the scales
on which @xmath117 is varying , we can evaluate the integrand of eq .
( [ e : naive2 ] ) at its central value @xmath118 and extract it from the integral .
the integral over the annulus @xmath111 cancels the factor @xmath119 in the denominator , reducing eq .
( [ e : naive2 ] ) to @xmath120 \ , .\ ] ] @xmath121 is indeed an estimator for the lensing - potential power spectrum @xmath106 , albeit a biaed one .
note that the _ bias _ in the power - spectrum estimator @xmath121 is precisely the same as the _ variance _ shown in eq .
( [ e : prodave ] ) with which we were able to determine each individual lensing mode .
this is no coincidence ; it reflects the fact that there are no grounds _ a priori _ on which to differentiate the variance with which we can reconstruct individual modes @xmath89 from the intrinsic variance @xmath122 of the underlying distribution from which they are drawn . to obtain an unbiased estimator to compare with theoretical predictions , we subtract off this unwanted reconstruction variance , @xmath123 \ , .\ ] ] since @xmath87 as defined in eq .
( [ e:1st ] ) itself depends on @xmath106 , this subtraction and evaluation must be performed iteratively until a self - consistent solution is obtained .
the variance of our estimator @xmath124 can be calculated in the usual manner , @xmath125 evaluating this expression requires us to calculate @xmath126 \left [ { { \mathbf{d}}_{\theta\theta}}({{{\bf l}_2 } } ' ) \cdot { { \mathbf{d}}_{\theta\theta}}(-{{{\bf l}_2 } } ' ) \right ] \rangle \ , .\ ] ] since @xmath58 is a quadratic estimator in the temperature map , eq .
( [ e : dl2 ] ) includes the following integral over the eight - point correlation function in fourier space , @xmath127 \left [ { { \mathbf{d}}_{\theta\theta}}({{{\bf l}_2 } } ' ) \cdot { { \mathbf{d}}_{\theta\theta}}(-{{{\bf l}_2 } } ' ) \right ] \rangle = \frac{{a_{\theta\theta}}^2(l_{1}^{\prime } ) { a_{\theta\theta}}^2(l_{2}^{\prime})}{(l_{1}^{\prime})^2 ( l_{2}^{\prime})^2 } { \int \frac{d^2 { \bf k}_1}{(2\pi)^2 } } { \int \frac{d^2 { \bf k}_3}{(2\pi)^2 } } { \int \frac{d^2 { \bf k}_5}{(2\pi)^2 } } { \int \frac{d^2 { \bf k}_7}{(2\pi)^2 } } \nonumber \\ & & \quad \times \big\ { { f_{\theta\theta}}({{{\bf k}_1 } } , { { { \bf k}_2 } } ) { f_{\theta\theta}}({{{\bf k}_3 } } , { { { \bf k}_4 } } ) { f_{\theta\theta}}({{{\bf k}_5 } } , { { { \bf k}_6 } } ) { f_{\theta\theta}}({{{\bf k}_7 } } , { { { \bf k}_8 } } ) \langle { \theta}^{{\rm t}}({{{\bf k}_1 } } ) { \theta}^{{\rm t}}({{{\bf k}_2 } } ) { \theta}^{{\rm t}}({{{\bf k}_3 } } ) { \theta}^{{\rm t}}({{{\bf k}_4 } } ) { \theta}^{{\rm t}}({{{\bf k}_5 } } ) { \theta}^{{\rm t}}({{{\bf k}_6 } } ) { \theta}^{{\rm t}}({{{\bf k}_7 } } ) { \theta}^{{\rm t}}({{{\bf k}_8 } } ) \rangle \big\ } \
, , \nonumber \\\end{aligned}\ ] ] where @xmath128 , @xmath129 , @xmath130 , and @xmath131 .
a fully general eight - point correlation function consists of a connected part , as well as terms proportional to the product of lower - order correlation functions . under the assumption that both the primordial cmb and the lensing potential are governed by gaussian statistics , all correlation functions higher than the four - point have vanishing connected parts @xcite .
the temperature eight - point correlation function will therefore be composed of three groups of terms ; membership in a group being determined by whether the term contains zero , one , or two factors of the trispectrum . since the trispectrum given in eq .
( [ e : trilens ] ) is first order in the lensing - potential power spectrum @xmath106 , terms of these three groups are zeroth , first , and second order respectively in @xmath106 .
combinatorics determines the number of terms in each group .
there are : @xmath132 different ways of dividing @xmath133 into four pairs , and hence there will be 105 terms in the group containing no trispectra .
similar calculations reveal that there are @xmath134 terms in the second group and @xmath135 terms in the third group .
many terms in all three groups will vanish for the same reason that the first term vanished in the four - point correlation function of eq .
( [ e:4pt ] ) ; these terms are proportional to a dirac delta function evaluated at nonzero argument .
consider now the first group of terms , those that are zeroth order in @xmath106 .
the 60 nonvanishing terms in this group each contain four dirac delta functions ; they can be further segregated into the 12 terms that allow two of the integrals over @xmath136 appearing in eq .
( [ e:8pt ] ) to be immediately evaluated via dirac delta functions , and the 48 terms that allow evaluation of three @xmath136 integrals .
the first 12 terms , inserted into eq .
( [ e:8pt ] ) and appropriately evaluated using the normalization of eq .
( [ e : norm ] ) , yield @xmath137 + \ldots \ , , \ ] ] while the remaining 48 terms give the final result to zeroth order in @xmath106 , @xmath138 \nonumber \\ & & \quad \quad \quad \quad \quad \quad + \frac{(2 \pi)^2}{a l^4 } \frac{2}{(2 \pi l \delta l)^2 } \int_{a_l } \frac{d^2 { { { \bf l}_1}}'}{(2 \pi)^2 } \ , \int_{a_l } \frac{d^2 { { { \bf l}_2}}'}{(2 \pi)^2 } \ , \frac{{a_{\theta\theta}}^2(l_{1}^{\prime } ) { a_{\theta\theta}}^2(l_{2}^{\prime})}{(l_{1}^{\prime})^2 ( l_{2}^{\prime})^2 } { \int \frac{d^2 { \bf k}_1}{(2\pi)^2 } } f_{{\theta}{\theta}}({{{\bf k}_1 } } , { { { \bf k}_2 } } ) p({{{\bf k}_1 } } , { { { \bf k}_2 } } , { { { \bf l}_1 } } ' , { { { \bf l}_2 } } ' ) \ , , \nonumber \\\end{aligned}\ ] ] where @xmath139 when @xmath140 is subtracted from @xmath141 in eq .
( [ e : sigpv ] ) , the first term in the square brackets of eq .
( [ e : zeropv ] ) will be eliminated . minimizing the variance associated with this estimator then consists of making an optimal choice of @xmath142 .
the first noise term is proportional to @xmath143 . for a survey of area @xmath110
, @xmath144 is the specific area of an individual mode in @xmath145 space and @xmath146 is the area in @xmath145 space over which the power - spectrum estimator takes an average .
this ratio is therefore the inverse of the number of individual @xmath58 modes whose inverse variances are added to determine the inverse variance of @xmath124 .
it is obviously minimized by choosing @xmath147 . the second term , that involving @xmath148 , differs from the first noise term in that a dirac delta function has been used to evaluate an additional @xmath136 integral rather than an annulus integral .
since the integrands are of the same order , we expect the second noise term to be suppressed relative to the first by a factor @xmath149 where @xmath150 is set by the resolution @xmath32 of the survey . under the conservative assumption @xmath151 , namely that we are probing scales well above our resolution , this term is assured to be small .
we neglect such terms for the remainder of this paper . if we insert the portions of the eight - point correlation function that are first and second order in @xmath106 into eq .
( [ e:8pt ] ) and evaluate using eq .
( [ e:1st ] ) , we find @xmath152 \ , , \ ] ] and @xmath153 this result agrees with that given in ref .
@xcite after subtracting our newly derived term @xmath87 .
the term @xmath87 and corresponding terms for polarization - based estimators have two principal effects on power spectrum estimation . as shown in eq .
( [ e : finvar ] ) , they provide a fractional contribution to the variance of roughly @xmath154 when @xmath122 dominates the variance as in the reference experiment in the right - hand panel of fig .
[ f : plrec ] . for the @xmath0cdm cosmological model considered here
this represents an increase of @xmath155 in the variance of the @xmath98 estimator for @xmath156 .
more importantly , @xmath87 acts as a bias for the naive estimator @xmath121 as shown by eq .
( [ e : naive3 ] ) . if this bias is not calculated and subtracted iteratively to form the unbiased estimator @xmath124 as in eq .
( [ e : pestsub ] ) , @xmath122 will be _ systematically _ overestimated by @xmath157 at low @xmath108 and by increasingly larger amounts at @xmath158 as the signal @xmath159 begins to plummet while @xmath87 remains comparatively flat .
having evaluated the variance of our estimator @xmath124 , we consider whether this estimator has a substantial covariance @xmath160 the estimator @xmath121 as defined in eq .
( [ e : naive ] ) implies that : @xmath161 \left [ { { \mathbf{d}}_{\theta\theta}}({{{\bf l}_2 } } ' ) \cdot { { \mathbf{d}}_{\theta\theta}}(-{{{\bf l}_2 } } ' ) \right ] \rangle \ , , \ ] ] which can be evaluated using the same integral over the eight - point correlation function described in eq .
( [ e:8pt ] ) . whereas 60 of the 105 zeroth - order terms in @xmath106 coming from this equation were nonvanishing for the variance , for the covariance only 52 terms are nonzero provided that the widths @xmath112 and @xmath162 are chosen so that the annuli @xmath111 and @xmath163 do not overlap .
this leads to a result analogous to eq .
( [ e : zeropv ] ) , @xmath164 note that the eight terms missing from the covariance when compared to the variance have altered the first term of eq .
( [ e : zerocv ] ) , and that it was precisely these terms that provided the dominant contribution to the variance of eq .
( [ e : finvar ] ) when @xmath142 was chosen appropriately .
we therefore find that to zeroth order in @xmath106 , the covariance is given by @xmath165 where we have extracted the @xmath166 from the annular integrals since they are slowly varying over the widths @xmath112 and @xmath162 . for the same reasons that terms of this form were a subdominant contribution to the variance as discussed previously , we expect the covariance to be suppressed as well . if we define the ratio @xmath167 we can quantify this suppression .
the triple integral of eq .
( [ e : fincov ] ) , appearing in the numerator of @xmath168 , involves integration over annuli with radii @xmath108 and @xmath169 and one integration over all fourier space .
the triple integrals in the variances @xmath170 and @xmath171 appearing in the denominator of @xmath168 each consist of a single integration over an annulus of radius @xmath108 and @xmath169 respectively and two integrations over all fourier space .
if we make the crude assumption that the integrand is constant , the ratio @xmath168 will simply be the ratio of these areas , @xmath172 the ratio @xmath168 is evaluated numerically for planck in fig .
[ f : cov ] as a function of @xmath108 for various fixed values of @xmath169 .
the estimators @xmath124 and @xmath173 were chosen such that @xmath174 , while integrals over fourier space were cut off at @xmath175 . substituting these values into eq .
( [ e : rcovest ] ) , we expect @xmath176 .
this crude estimate is surprisingly close to the numerically obtained results of fig .
[ f : cov ] ; in particular the slope of the curves is approximately 1/2 on this log - log plot . even at @xmath101 , @xmath177 suggesting that covariance in power - spectrum estimation
can safely be neglected for planck .
the estimate of eq .
( [ e : rcovest ] ) implicitly depends on the experimental resolution @xmath32 because the integrand appearing in expressions for the variance and covariance decreases rapidly for @xmath178 . for future experiments with better resolution than planck
, @xmath179 will be higher implying by eq .
( [ e : rcovest ] ) that covariance will be even more negligible .
weak gravitational lensing induces non - gaussian correlations between modes of the observed cmb temperature map as shown in eq .
( [ e : ea ] ) . these correlations , and assumptions about the gaussian nature of the primordial cmb , can be used to construct several temperature and polarization - based estimators of the fourier modes @xmath89 of the deflection field .
this procedure was outlined in ref .
@xcite , however in calculating the noise associated with this reconstruction , an assumption was made that the observed temperature map was gaussian . in the presence of lensing this assumption
is invalid ; when calculating the variance of quadratic estimators all permutations of the observed trispectrum must be taken into account .
one such permutation reflects the desired correlation making our estimator sensitive to @xmath89 , but the remaining two permutations induce additional variance proportional to the lensing - potential power spectrum @xmath106 . while subdominant , this variance will become increasing significant for future experiments as shown in the right - hand panel of fig .
[ f : plrec ] .
since the power spectrum @xmath106 is itself a measure of uncertainty in the deflection field , this additional variance in lensing reconstruction acts as a bias during power - spectrum estimation because there is no _ a priori _ way to distinguish it from the intrinsic variance of the underlying distribution .
our calculation of the dependence of this variance on @xmath106 allows it in principle to be subtracted iteratively , which will prevent a systematic @xmath157 overestimate of @xmath106 at low @xmath108 .
we close by considering several possible observational obstacles to the scheme for lensing reconstruction and power - spectrum estimation presented above .
one hindrance is secondary contributions to the cmb such as the sz and isw effects .
these effects increase the total temperature power spectrum appearing in the denominator of the optimum filter @xmath104 of eq .
( [ e : filt ] ) as would additional instrumental noise .
they also correlate with the large - scale structure at low redshifts inducing further non - gaussian couplings and additional variance to lensing reconstruction .
fortunately for our purposes the frequency dependence of the thermal sz effect differs from that of a blackbody .
it can therefore be separated in principle from the lensed primordial cmb by an experiment with several frequency channels @xcite .
the isw effect can not be removed in this manner , but is too small to significantly inhibit lensing reconstruction .
polarization - dependent secondary effects are expected to appear at higher orders in the density contrast @xcite , and we therefore anticipate that they will not make a contribution at the levels considered here . a
potentially more serious problem is that of galactic foregrounds , which though uncorrelated with the lensing signal may be substantial at certain frequencies .
significant polarization has also been observed in some of these sources @xcite .
we hope to understand and minimize the effects of galactic foregrounds in future work , and to pursue further refinements of lensing reconstruction .
we thank wayne hu for useful discussions .
this work was supported in part by nasa nag5 - 11985 and doe de - fg03 - 92-er40701 .
kesden acknowledges the support of an nsf graduate fellowship and ac acknowledges support from the sherman fairchild foundation .
here we provide the appropriate formulas for deriving the variance associated with polarization - based estimators of the deflection field @xmath89 .
the cmb polarization can be decomposed into e and b - modes @xcite .
these modes are mixed by weak lensing such that to linear order in @xmath68 , @xmath180 \phi({{\bf l}}-{{\bf l}}_1 ) \left [ ( { { \bf l}}-{{\bf l}}_1 ) \cdot { { \bf l}}_1 \right ] \
, , \nonumber \\
\tilde b({{\bf l } } ) & = & b({{\bf
l } } ) - { \int \frac{d^2 { \bf l}_1}{(2\pi)^2 } } \left[e({{\bf l}}_1 ) \sin 2 ( \varphi_{{{\bf l}}_1 } -\varphi_{{\bf l } } ) + b({{\bf l}}_1 ) \cos 2 ( \varphi_{{{\bf l}}_1 } -\varphi_{{\bf l}})\right ] \phi({{\bf l}}-{{\bf
l}}_1 ) \left [ ( { { \bf l}}-{{\bf l}}_1 ) \cdot { { \bf l}}_1 \right ] \ , .\end{aligned}\ ] ] we can exploit the sensitivity of the polarization modes to the lensing potential to construct lensing estimators from quadratic combinations of polarization modes .
we generalize eq .
( [ e : ea ] ) to arbitrary combinations @xmath181 of @xmath45 , e , and b - modes as first derived in ref .
@xcite , @xmath182 where @xmath183 \cos 2 ( \varphi_{{{\mathbf{l } } } } -\varphi_{{\mathbf{l^{\prime } } } } ) \ , , \nonumber \\
f_{eb}({{\mathbf{l } } } , { { \mathbf{l^{\prime } } } } ) & = & \left [ c_{l}^{ee } ( { { \mathbf{l}}}\cdot { { \mathbf{l } } } ) + c_{l^{\prime}}^{bb } ( { { \mathbf{l}}}\cdot { { \mathbf{l^{\prime } } } } ) \right ] \sin 2 ( \varphi_{{{\mathbf{l } } } } -\varphi_{{\mathbf{l^{\prime } } } } ) = f_{be}({{\mathbf{l^{\prime } } } } , { { \mathbf{l } } } ) \ , , \nonumber \\ f_{bb}({{\mathbf{l } } } , { { \mathbf{l^{\prime } } } } ) & = & \left [ c_{l}^{bb } ( { { \mathbf{l}}}\cdot { { \mathbf{l } } } ) + c_{l^{\prime}}^{bb } ( { { \mathbf{l}}}\cdot { { \mathbf{l^{\prime } } } } ) \right ] \cos 2 ( \varphi_{{{\mathbf{l } } } } -\varphi_{{\mathbf{l^{\prime } } } } ) \ , . \nonumber \\\end{aligned}\ ] ] in deriving these results we used parity considerations to demand @xmath184 . using these relations
, we follow the approach of ref .
@xcite to derive symmetric lensing estimators @xmath185 and @xmath186 { f_{xx'}}({{{\bf l}_1 } } , { { { \bf l}_2 } } ) \ , .\ ] ] we have explicitly symmetrized our estimators for @xmath187 to simplify the form of the optimal filter .
the normalization bias of the estimators is removed by choosing @xmath188^{-1 } \
, , \ ] ] and @xmath189 { f_{xx'}}({{{\bf l}_1 } } , { { { \bf l}_2 } } ) \right]^{-1 } \ , .\ ] ] the minimum - variance filters @xmath190 for the various cases @xmath181 are given by @xmath191 using these optimal filters for the estimators defined in eqs .
( [ e : estxx ] ) and ( [ e : estxx ] ) , we can calculate the variances for these estimators in a fashion entirely analogoug to eq .
( [ e : estvar ] ) , @xmath192 @xmath193 \left [ x^{{\rm t}}({{{\bf l}_1 } } ' ) x'^{{\rm t}}({{{\bf l}_2 } } ' ) + x'^{{\rm t}}({{{\bf l}_1 } } ' ) x^{{\rm t}}({{{\bf l}_2 } } ' ) \right ] \rangle { f_{xx'}}({{{\bf l}_1 } } , { { { \bf l}_2 } } ) { f_{xx'}}({{{\bf l}_1 } } ' , { { { \bf l}_2 } } ' ) \nonumber \\ & & \quad \quad - ( 2 \pi)^2 \delta_{{\rm d}}({{\mathbf{l}}}- { { \mathbf{l^{\prime } } } } ) c_{l}^{dd } \ , . \nonumber \\\end{aligned}\ ] ] as for that of the temperature estimator , these variance will consist of zeroth - order terms in @xmath106 , @xmath194 and @xmath195 , and first - order terms , @xmath196 @xmath197 \nonumber \\ & & \quad \quad \quad + c_{|{{{\bf l}_1}}-{{{\bf l}_2}}'|}^{\phi\phi } \bigl [ f_{xx'}(-{{{\bf l}_1 } } , { { { \bf l}_2 } } ' ) f_{x'x}(-{{{\bf l}_2 } } , { { { \bf l}_1 } } ' ) + f_{xx}(-{{{\bf l}_1 } } , { { { \bf l}_2 } } ' ) f_{x'x'}(-{{{\bf l}_2 } } , { { { \bf l}_1 } } ' ) \nonumber \\ & & \quad \quad \quad
\quad + f_{x'x'}(-{{{\bf l}_1 } } , { { { \bf l}_2 } } ' ) f_{xx}(-{{{\bf l}_2 } } , { { { \bf l}_1 } } ' ) + f_{x'x}(-{{{\bf l}_1 } } , { { { \bf l}_2 } } ' ) f_{xx'}(-{{{\bf l}_2 } } , { { { \bf l}_1 } } ' ) \bigl ] \big\ } \ , , \nonumber \\\end{aligned}\ ] ] the filters given in eq . (
[ e : filtxx ] ) are no longer optimal in the presence of this additional noise , but the difference between these filters and the optimal filters should be negligible provided that @xmath198 , @xmath199 . for the purposes of power - spectrum estimation , the terms @xmath200 and @xmath201 are not only an additional contribution to the variance , but are also a systematic bias if not subtracted iteratively following eq .
( [ e : naive3 ] ) .
a final point to consider is that the six different estimators @xmath202 defined in this paper are not independent , as they are constructed from only three distinct maps .
the covariance matrix for the six estimators will therefore not be diagonal , and this needs to be taken into account if the estimators are to be linearly combined to produce a single minimum - variance estimator .
the off - diagonal elements of the covariance matrix can be evaluated in a straightforward manner involving pairs of double integrals similar to those of eqs .
( [ e : xxvar ] ) and ( [ e : xxvar ] ) . p. j. e. peebles and j. t. yu , astrophys . j. * 162 * , 815 ( 1970 ) ; r. a. sunyaev and ya . b. zeldovich , astrophys .
space sci .
* 7 * , 3 ( 1970 ) ; j. silk , astrophys .
j. * 151 * , 459 ( 1968 ) ; w. hu and s. dodelson , ann .
astrophys . * 40 * , 171 ( 2002 ) .
g. jungman _
et al . _ ,
d * 54 * , 1332 ( 1995 ) ; j. r. bond , g. efstathiou , and m. tegmark , mon . not .
r. astron .
soc . * 291 * , l33 ( 1997 ) ; m. zaldarriaga and d. n. spergel , astrophys . j. * 488 * , 1 ( 1997 ) ; d. j. eisenstein , w. hu , and m. tegmark , astrophys . j. * 518 * , 2 ( 1999 ) .
d. n. spergel and d. m. goldberg , phys .
d * 59 * , 103001 ( 1999 ) ; d. m. goldberg and d. n. spergel , phys . rev . d * 59 * , 103002 ( 1999 ) ; a. cooray and w. hu , astrophys .
j. * 534 * , 533 ( 2000 ) ; m. zaldarriaga and u. seljak , phys .
rev . d * 59 * , 123507 ( 1999 ) ; h. v. peiris and d. n. spergel , astrophys . j. * 540 * , 605 ( 2000 ) .
g. giardino _
et al . _ ,
astrophys .
* 387 * , 82 ( 2002 ) ; f. r. bouchet , s. prunet , and s. k. sethi , mon . not .
r. astron .
302 * , 663 ( 1999 ) ; s. prunet , s. k. sethi , and f. r. bouchet , mon . not .
r. astron .
314 * , 348 ( 2000 ) . | weak gravitational lensing by intervening large - scale structure induces a distinct signature in the cosmic microwave background ( cmb ) that can be used to reconstruct the weak - lensing displacement map .
estimators for individual fourier modes of this map can be combined to produce an estimator for the lensing - potenial power spectrum .
the naive estimator for this quantity will be biased upwards by the uncertainty associated with reconstructing individual modes ; we present an iterative scheme for removing this bias .
the variance and covariance of the lensing - potenial power spectrum estimator are calculated and evaluated numerically in a @xmath0cdm universe for planck and future polarization - sensitive cmb experiments . | [
2,
109,
20687,
7270,
1688,
143,
4761,
1271,
110,
158,
140,
3943,
173,
59373,
211,
718,
56738,
135,
109,
1290,
415,
23313,
5651,
173,
109,
5447,
140,
209,
31187,
231,
459,
110,
107,
106,
109,
3404,
2198,
113,
219,
59373,
905,
1596,
84003,
316,
112,
109,
799,
242,
110,
108,
111,
556,
113,
153,
25020,
484,
5782,
253,
130,
9596,
12932,
111,
109,
47996,
6519,
2650,
757,
46819,
1093,
2324,
257,
160,
83278,
5384,
110,
107,
10957,
120,
109,
66723,
4761,
1271,
117,
110,
45350,
41403,
110,
108,
609,
233,
110,
45350,
41403,
44058,
115,
109,
5221,
1834,
137,
129,
263,
112,
37003,
109,
43983,
423,
233,
2116,
1557,
2650,
757,
46819,
110,
107,
115,
136,
800,
106,
110,
108,
145,
1693,
109,
575,
113,
79145,
109,
5404,
233,
4022,
273,
866,
135,
142,
5221,
4761,
1271,
1972,
1834,
110,
107,
106,
145,
403,
120,
136,
13187,
7241,
134,
11169,
241,
186,
117,
4414,
484,
110,
108,
118,
109,
310,
870,
120,
189,
440,
5083,
192,
2516,
109,
6254,
113,
189,
3846,
120,
117,
799,
134,
360,
11169,
110,
107,
106,
209,
141,
60252,
136,
3027,
233,
2531,
39155,
137,
156,
76336,
6595,
109,
66723,
3027,
233,
8190,
162,
1278,
130,
114,
861,
233,
1991,
3846,
113,
77076,
69369,
116,
2650,
757,
46819,
110,
107,
1,
0,
0,
0,
0,
0,
0,
0,
0,
0,
0,
0,
0,
0,
0,
0,
0,
0,
0,
0,
0,
0,
0,
0,
0,
0,
0,
0,
0,
0,
0,
0,
0,
0,
0,
0,
0,
0,
0,
0,
0
] |
the study of the equation of state ( eos ) , transport properties , and mixing rules of hydrogen ( h ) and helium ( he ) under extreme condition of high pressure and temperature is not only of fundamental interest but also of essential practical applications for astrophysics @xcite .
for instance , giant planets such as jupiter and saturn require accurate eos as the basic input into the respective interior models in order to solve hydrostatic equation and investigate the solubility of the rocky core @xcite . on the other side , the evolution of stars and the design of thermal protection system is assisted by high precision transport coefficients of h - he mixtures at high pressure @xcite . in addition ,
the viscosity and mutual diffusion coefficients are also important input properties for hydrodynamic simulations in modelling the stability of the hot spot - fuel interfaces and the degree of fuel contamination in inertial confinement fusion ( icf ) @xcite . since direct experimental access such as shock wave experiments
is limited in the mbar regime @xcite , the states deep in the interior of jupiter ( @xmath2 mbar ) and saturn ( @xmath3 mbar ) @xcite can not be duplicated in the laboratory . as a consequence ,
theoretical modelling provides most of the insight into the internal structure of giant planets .
the eos of h - he mixtures have been treated by a linear mixing ( lm ) of the individual eos via fluid perturbation theory @xcite and monte carlo simulations @xcite . recently , several attempts have been made to calculate eos of h - he mixtures by means of quantum molecular dynamic ( qmd ) simulations .
@xcite applied local density approximation of density functional theory ( lda - dft ) calculations for solid h - he mixtures , implying demixing for jupiter and saturn at 15000 k for a he fraction of @xmath4 .
@xcite , lorenzen _ et al . _
@xcite , and militzer @xcite performed qmd simulations by using generalized gradient expansion ( gga ) instead of the lda in order to evaluate the accuracy of the lm approximation and study the demixing of h - he at mbar pressures .
@xcite introduced car - parrinello molecular dynamics ( cpmd ) simulations to calculate the excess gibbs free energy of mixing at a lower temperature compared to that set in the work of lorenzen _ et al .
_ @xcite and militzer @xcite . considering the transport properties , qmd @xcite and orbital - free molecular dynamics ( ofmd ) @xcite simulations
have been introduced to study hydrogen and its isotropic deuterium ( d ) and tritium ( t ) .
self - diffusion coefficients in the pure h system and mutual diffusion for d - t mixtures were determined for temperatures @xmath51 to 10 ev and equivalent h mass densities 0.1 to 8.0 g / cm@xmath1 @xcite .
qmd and ofmd simulations of self - diffusion , mutual diffusion , and viscosity have recently been performed on heavier elements ( fe , au , be ) @xcite and on mixtures of li and h @xcite .
the present work selects h - he mixture as a representative system and examines some of the standard mixing rules with respect to the eos and transport properties ( viscosity , self and mutual diffusion coefficients ) in the warm dense regime that covers standard extreme condition as reached in the interiors of jupiter and saturn .
the thermophysical properties of the full mixture and the individual species have been derived from qmd simulations , where the electrons are quantum mechanically treated through finite - temperature ( ft ) dft and ions move classically . in the next section , we present the formalism for qmd and for determining the static and transport properties .
then , eos , viscosity , and diffusion coefficients for h - he mixtures are presented , and the qmd results are compared with the results from reduced models and qmd based linear mixing models . finally , concluding remarks are given .
in this section , a brief description of the fundamental formalism employed to investigate h - he mixtures is introduced .
the basic quantum mechanical density functional theory forms the basis of our simulations .
the implementation of schemes in determining diffusion and viscosity is discussed . mixing rules that combine pure species quantities to form composite properties
is also presented .
qmd simulations have been performed for h - he mixtures by using vienna _
ab initio _ simulation package ( vasp ) @xcite . in these simulations , the electrons are treated fully quantum mechanically by employing a plane - wave ft - dft description , where the electronic states follow the fermi - dirac distribution .
the ions move classically according to the forces from the electron density and the ion - ion repulsion .
simulations have been performed in the nvt ( canonical ) ensemble where the number of particles @xmath7 and the volume are fixed .
the system was assumed to be in local thermodynamic equilibrium with the electron and ion temperatures being equal ( @xmath8 ) . in these calculations ,
the electronic temperature has been kept constant according to fermi - dirac distribution , and ion temperature is controlled by noe thermostat @xcite . at each step during md simulations , a set of electronic state functions [ @xmath9 for each * k*-point
are determined within kohn - sham construction by @xmath10 with @xmath11 in which the four terms respectively represent the kinetic contribution , the electron - ion interaction , the hartree contribution and the exchange - correlation term . the electronic density is obtained by @xmath12 then by applying the velocity verlet algorithm , based on the force from interactions between ions and electrons , a new set of positions and velocities are obtained for ions .
all simulations are performed with 256 atoms and 128 atoms for pure species of h and he , and as for the case of the h - he mixture , a total number of 245 atoms ( 234 h atoms and 11 he atoms ) for a mixing ratio @xmath13 ( corresponding to the h - he immiscibility region determined in the work of morales _ et al . _
@xcite ) has been adopted , where a cubic cell of length @xmath14 ( volume @xmath15 ) is periodically repeated .
the simulated densities range from 1.0 to 4.0 g / cm@xmath1 for pure h system . as for pure
he and h - he mixture , the size of the supercell is chosen to be the same as that for pure h to secure a constant electron number density ( in the range @xmath0/m@xmath1 ) . the temperature from 4000 k to 20000 k
has been selected to highlight the conditions in the interiors of jupiter and saturn .
the convergence of the thermodynamic quantities plays an important role in the accuracy of qmd simulations . in the present work ,
a plane - wave cutoff energy of 1200 ev is employed in all simulations so that the pressure is converged within 2% .
we have also checked out the convergence with respect to a systematic enlargement of the * k*-point set in the representation of the brillouin zone . in the molecular dynamic simulations ,
only the @xmath16 point of brillouin zone is included .
the dynamic simulation is lasted 20000 steps with time steps of 0.2 @xmath17 0.7 fs according to different densities and temperatures . for each pressure and temperature ,
the system is equilibrated within 0.5 @xmath17 1 ps .
the eos data are obtained by averaging over the final 1 @xmath17 3 ps molecular dynamic simulations .
the self - diffusion coefficient @xmath18 can either be calculated from the trajectory by the mean - square displacement @xmath19 or by the velocity autocorrelation function @xmath20 where @xmath21 is the position and @xmath22 is the velocity of the @xmath23th nucleus . only in the long - time limit , these two formulas of @xmath18 are formally equivalent .
sufficient lengths of the trajectories have been generated to secure contributions from the velocity autocorrelation function to the integral is zero , and the mean mean - square displacement away from the origin consistently fits to a straight line .
the diffusion coefficient obtained from these two approaches lie within 1 % accuracy of each other .
here , we report the results from velocity autocorrelation function .
we have also computed the mutual - diffusion coefficient @xmath24 from the autocorrelation function @xmath25 with @xmath26 where the concentration and particle number of species @xmath27 are denoted by @xmath28 and @xmath29 , respectively , and the total number of particles in the simulation box @xmath30 .
the quantity @xmath31 is the thermodynamic factor related to the second derivation of the gibbs free energy with respect to concentrations @xcite . in the present simulations ,
@xmath31 value has been adopted equal to unity since studies with leonard - jones and other model potentials have shown that for dissimilar constituents the @xmath31-factor departs from unity by about 10% @xcite .
the viscosity @xmath32 has been computed from the autocorrelation function of the off - diagonal component of the stress tensor @xcite @xmath33 the results are averaged from the five independent off - diagonal components of the stress tensor @xmath34 , @xmath35 , @xmath36 , @xmath37 , and @xmath38 .
different from the self - diffusion coefficient , which involves single - particle correlations and attains significant statistical improvement from averaging over the particles , the viscosity depends on the entire system and thus needs very long trajectories so as to gain statistical accuracy . to shorten the length of the trajectory , we use empirical fits @xcite to the integrals of the autocorrelation functions .
thus , extrapolation of the fits to @xmath39 can more effectively determine the basic dynamical properties .
both of the @xmath18 and @xmath40 have been fit to the functional in the form of @xmath41 $ ] , where @xmath42 and @xmath43 are free parameters .
reasonable approximation to the viscosity can be produced from the finite time fitting procedure , which also serves to damp the long - time fluctuations .
the fractional statistical error in calculating a correlation function @xmath44 for molecular - dynamics trajectories @xcite can be given by @xmath45 where @xmath43 is the correlation time of the function , and @xmath46 is the length of the trajectory .
in the present work , we generally fitted over a time interval of [ 0 , @xmath47 . here
, we examine two representative mixing rules .
the first , termed density - matching rule ( mrd ) with the inspiration of a two - species ideal gas .
the second , termed pressure - matching rule ( mrp ) , which follows from two interacting immiscible fluids . in the mrd
, the volume of the individual species is set equal to that of the mixture ( @xmath48 ) , and qmd simulations are performed for h at a density of @xmath49 and he at @xmath50 at a temperature @xmath51 .
then , pressure predicted by mrd is determined by simply adding the individual pressures from the pure species h and he simulations .
other transport coefficients , such as mutual diffusion and viscosity , follow the same prescription and are summarized as @xmath52 the superscript is used to denote values predicted from mrd .
the derived pressure based on density mixing rule generally follows from the ideal noninteracting h and he gas in a volume @xmath53 .
the mrp has a more complicated construction compared to mrd .
mrp can be characterized as the following prescription : @xmath54 in this case , we have performed a series of qmd simulations on the individual species h and he , where the volumes change under a constraint @xmath55 until the individual pressures equal to each other @xmath56 .
the total pressure becomes the predicted value .
here , we use the excess or electronic pressure @xmath57 to evaluate this mrp mixing rule .
composite properties such as mutual diffusion and viscosity are evaluated by combining the individual species results via volume fractions @xmath58 .
finally , we also derive properties of the mixture from a slightly more complex mixing rule @xcite , as so - called binary ionic mixture ( bim ) : @xmath59 with @xmath60 the predicted mutual - diffusion coefficient or the viscosity .
the subscript @xmath61 denotes the mixture and @xmath23 the pure species .
in this section , the wealth of information derived from qmd calculations are mainly presented through figures , and the general trends of the eos as well as transport coefficients are concentrated in the text .
it is , therefore , interesting to explore not only to get insight into the interior physical properties of giant gas planets but also to examine a series of mixing rules for hydrogen and helium .
additionally , one can consider the influence of helium on the eos and transport coefficients of mixing .
high precision eos data of hydrogen and helium are essential for understanding the evolution of jupiter and target implosion in icf .
experimentally , the eos of hydrogen and helium in the fluid regime have been studied through gas gun @xcite , chemical explosive @xcite , magnetic driven plate flyer @xcite , and high power laser @xcite . since these experiments were limited by the conservation of mass , momentum , and energy , the explored density of warm dense matter
were limited within @xmath62 times of the initial density .
recently , a new technique combined diamond anvil cell ( dac ) and high intensity laser pulse has successfully been proved to provide visible ways to generate shock huguniot data of hydrogen over a significantly broader density - temperature regime than previous experiments @xcite
. however , the density therein was still restricted within 1 g / cm@xmath1 . in our simulations , wide range eos for h
, he , and h - he mixtures have been determined according to qmd method .
the eos can be divided into two parts that is , contributions from the noninteracting motion of ions ( @xmath63 ) and the electronic term ( @xmath64 ) , @xmath65 where @xmath64 is calculated directly through dft . in fig .
[ fig_eos1 ] ( a ) , we have compared our results of @xmath64 with that of holst _ et al . _
@xcite , where the electronic pressure is expressed as a smooth function in terms of density and temperature , and the results agree with each other with a very slight difference ( accuracy within 5% ) . in the simulated density and temperature regime
, we do not find any signs indicating a liquid - liquid phase transition ( @xmath66 ) or plasma phase transition ( @xmath67 ) , which are characterized by molecular dissociation and ionization of electrons , respectively . with considering the mixing of he into h ,
the electronic pressure is effectively reduced , as has been shown in fig .
[ fig_eos1 ] ( b ) .
mrd accounts for contribution from noninteracting h and he subsystems in the volume of the mixtures . in mrd ,
the pressure contributed from noninteracting ions is the same as that of the mixture , but the electronic pressure is much lower due to the low electronic density in the pure species simulations .
it is indicated that the electronic pressure is underestimated by mrd model at about 8% @xmath17 9% ( see fig .
[ fig_eos2 ] ) . for mrp model
, we have firstly performed a series of pure species simulations at a wider density ( temperature ) regime compared to h - he mixtures .
then , the simulated eos data are fitted into smooth functions in terms of density and temperature . under the constraint of @xmath68
, we have predicted the electronic pressures @xmath69 according to mrp model by solving pure species eos function at certain densities and temperatures , as shown in fig .
[ fig_eos2 ] .
it is indicated that the mrp model agrees better with direct qmd simulations ( accuracy within 3% ) , the difference mainly come from the ionic interactions between h and he species after mixing . and 8 g / cm@xmath1 , respectively .
( a ) self - diffusion coefficient ; ( b ) viscosity .
the direct qmd simulated results are presented by black crosses , while the fitted results are denoted by blue dashed lines . ]
qmd simulations have been performed within the framework of ft - dft to benchmark the dynamic properties of h , he , and h - he mixture in the wdm regime .
illustrations for the self - diffusion coefficients and viscosity ( for h and he at densities of 2.0 g / cm@xmath1 and 8.0 g / cm@xmath1 , respectively ) at a temperature 12000 k , as well as their fits are shown in fig . [ fig_dandeta ] .
the trajectory of the present simulations lasts 4.0 @xmath17 14.0 ps , and correlation times between 1.0 and 15.0 fs . as a consequence ,
the computational error for the viscosity lies within 10% .
after accounting for the fitting error and extrapolation to infinite time , a total uncertainty of @xmath17 20% can be estimated .
the uncertainty in the self - diffusion coefficients is smaller than 1% , due to the additional @xmath70 advantage given by particle average . .
for helium , the density is 8 g / cm@xmath1 .
the electron number density for pure species ( h and he ) is @xmath71/m@xmath1 .
] dynamic properties of wdm are generally governed by two dimensionless quantities , namely , ionic coupling ( @xmath16 ) and electronic degenerate ( @xmath72 ) parameter .
the former one is defined by the ratio of the potential to kinetic energy @xmath73 , with @xmath74 the ionic charge , and @xmath75 the ion - sphere radius ( @xmath76 is the number density ) . the latter one @xmath77 , where @xmath78 is fermi temperature .
it has been reported that dynamic properties such as diffusion coefficients and viscosity can be represented purely in terms of ionic coupling parameter @xmath16 according to molecular dynamics or monte carlo simulations based on one component plasma ( ocp ) model @xcite , where ions move classically in a neutralizing background of electrons .
for instance , hansen _
@xcite introduced a memory function to analyze the velocity autocorrelation function , and obtain the diffusion coefficient in terms of @xmath79 with the plasma frequency @xmath80 .
based on classical molecular dynamic simulations , bastea @xcite has fitted the viscosity into the following form @xmath81 with @xmath82 , @xmath83 , @xmath84 , and @xmath85 .
since ocp model is restricted to a fully ionized plasma , we use @xmath74=1.0 ( or 2.0 ) for hydrogen ( or helium ) to compute the self - diffusion coefficient and viscosity . in fig .
[ fig_qmd_ocp ] ( a ) , we show comparison between qmd and ocp model @xcite for hydrogen and helium at densities of 2 g / cm@xmath1 and 8 g / cm@xmath1 .
the general tendency for the self - diffusion coefficient with respect to temperature is similar for qmd and ocp model , however , the difference up to @xmath1760 % is observed between the two results . for the viscosity [ fig .
[ fig_qmd_ocp ] ( b ) ] , ocp @xcite predicts smaller ( larger ) values for hydrogen ( helium ) compared to qmd simulations .
the viscosity is governed by interactions between particles and ionic motions , contribution from the former one decrease with the increase of temperature , while , it increases for the latter one . as a consequence
, the viscosity may have local minimum along temperature . for hydrogen
, the local minimum locates around 10000 k and 14000 k indicated by qmd and ocp model , respectively . while in the case of helium
, we do not observe any signs for the local minimum in the simulated regime .
/m@xmath1 and temperature from 4000 @xmath17 20000 k. ] in fig .
[ fig_mix ] , we have shown the mutual diffusion coefficient @xmath86 and viscosity @xmath87 for h - he mixture with an electron number density of @xmath88/m@xmath1 , and results from mixing models are also provided .
the transport coefficients predicted by mrd can be directly evaluated through eq .
( [ eq_mrd ] ) . for mrp model ,
we have firstly fitted the self - diffusion coefficients and viscosity in terms of density and temperature , after determining the volume for each species under the constraint of @xmath89 , the transport coefficients are then obtained . in bim model , we have used @xmath90 and @xmath91 , then , the transport coefficients are determined by eq .
( [ eq_bim ] ) . here
we would like to stress that in some mixture studies based on average atom models , the properties of pure species are derived from perturbed - atom models , where boundary conditions are introduced from the surrounding medium by treating a single atom within a cell . in the present work ,
the dynamic properties of different mixing rules originate from qmd calculations of the individual species . despite divorced of the h -
he interactions , the pure species calculations still contain complex intra - atomic interactions based on large samples of atoms .
the mutual diffusion coefficient of h - he mixture shows a linear increase with respect of temperature , as indicated in fig .
[ fig_mix](a ) .
the data from mrp and bim models have a better agreement with qmd simulations compared with that of the mrd model , where ion densities are reduced and results in a larger diffusion coefficient .
the viscosity of h - he mixture has a more complex behavior than pure species under extreme condition . as shown in fig .
[ fig_mix ] ( b ) , mrd is valid at low temperature , while mrp works at higher temperature .
bim rule moves the results into better agreement with the h - he mixture , leaving within 30% or better for the simulated conditions .
in summary , we have performed systematic qmd simulations of h , he , and h - he mixture in the warm dense regime for electron number density ranging from @xmath0/m@xmath1 and for temperatures from 4000 to 20000 k. the present study concentrated on thermophysical properties such as the eos , diffusion coefficient , and viscosity , which are of crucial interest in astrophysics and icf .
various mixing rules have been introduced to predict dynamical properties from qmd simulations of the pure species and compare with direct calculations on the fully interacting mixture .
we have shown that mrd and mrp rules produce pressures within about 10 % of the h - he mixture , however , the mutual diffusion coefficients are as different as 75 % and it is 50 % for the viscosity .
bim rule generally gives better agreement with the mixture results .
we have also compared our qmd results with ocp model for the pure species .
this work was supported by nsfc under grants no .
11275032 , no . 11005012 and no . 51071032 , by the national basic security research program of china , and by the national high - tech icf committee of china .
99 w. b. hubbard , science * 214 * , 145 ( 1981 ) . h. f. wilson and b. militzer , astrophys .
j. * 745 * , 54 ( 2012 ) .
h. f. wilson and b. militzer , phys .
lett . * 108 * , 111101 ( 2012 ) .
d. bruno , c. catalfamo , m. capitelli , g. colonna , o. de pascale , p. diomede , c. gorse , a. laricchiuta , s. longo , d. giordano , and f. pirani , phys .
plasmas * 17 * , 112315 ( 2010 ) . s. atzeni and j. meyer - ter - vehn , _ the physics of inertial fusion : beam plasma interaction , hydrodynamics , hot dense matter _ , international series of monographs on physics ( clarendon press , oxford , 2004 ) . j. d. lindl , _ inertial confinement fusion : the quest for ignition and energy gain using indirect drive _ , ( springer - verlag , new york , 1998 ) .
m. d. knudson and m. p. desjarlais , phys .
lett . * 103 * , 225501 ( 2009 ) .
w. lorenzen , b. holst , and r. redmer , phys .
lett . * 102 * , 115701 ( 2009 ) .
d. j. stevenson , phys .
b * 12 * , 3999 ( 1975 ) .
w. b. hubbard and h. e. dewitt , astrophys .
j. * 290 * , 388 ( 1985 ) .
j. e. klepeis , k. j. schafer , t. w. barbee iii , and m. ross , science * 254 * , 986 ( 1991 ) .
j. vorberger , i. tamblyn , b. militzer , and s. a. bonev , phys .
b * 75 * , 024206 ( 2007 ) .
b. militzer , phys .
b * 87 * , 014202 ( 2013 ) .
o. pfaffenzeller , d. hohl , and p. ballone , phys .
lett . * 74 * , 2599 ( 1995 ) .
l. a. collins , j. d. kress , d. l. lynch , and n. troullier , j. quant .
transf . * 51 * , 65 ( 1994 ) .
i. kwon , j. d. kress , and l. a. collins , phys .
b * 50 * , 9118 ( 1994 ) .
l. collins , i. kwon , j. kress , n. troullier , and d. lynch , phys .
e * 52 * , 6202 ( 1995 ) .
j. clrouin and s. bernard , phys .
e * 56 * , 3534 ( 1997 ) .
i. kwon , l. collins , j. kress , and n. troullier , epl * 29 * , 537 ( 1995 ) .
j. clrouin and j .- f .
dufrche , phys .
e * 64 * , 066406 ( 2001 ) .
j. d. kress , j. s. cohen , d. a. horner , f. lambert , and l. a. collins , phys .
e * 82 * , 036404 ( 2010 ) .
f. lambert , j. clouin , and s. mazevet , epl * 75 * , 681 ( 2006 ) .
f. lambert , j. clouin , and g. zrah , phys .
e * 73 * , 016403 ( 2006 ) . c. wang , y. long , x .-
he , and p. zhang , phys .
e * 87 * , 043105 ( 2013 ) .
d. a. horner , f. lambert , j. d. kress , and l. a. collins , phys .
b * 80 * , 024305 ( 2009 ) .
m. a. morales , c. pierleoni , e. schwegler , and d. m. ceperley , proc .
* 106 * , 1324 ( 2009 ) .
j. p. hansen and i. r. mcdonald , _ the theory of simple liquids _ , 2nd ed .
( academic press , new york , 1986 ) , chap . 8 .
m. schoen and c. hoheisel , mol.phys .
* 52 * , 33 ( 1984 ) ; * 52 * , 1029 ( 1984 ) .
m. p. allen and d. j. tildesley , _ computer simulation of liquids _
( oxford university press , new york , 1987 ) . j. d. kress , james s. cohen , d. p. kilcrease , d. a. horner , and l. a. collins , phys .
e * 83 * , 026404 ( 2011 ) .
r. zwanzig and n. k. ailawadi , phys . rev . * 182 * , 280 ( 1969 ) .
s. bastea , phys .
e * 71 * , 056405 ( 2005 ) . w. j. nellis , rep .
* 69 * , 1479 ( 2006 ) .
g. v. boriskov , a. i. bykov , r. ilkaev , v. d. selemir , g. v. simakov , r. f. trunin , v. d. urlin , a. n. shuikin , and w. j. nellis , phys .
b * 71 * , 092104 ( 2005 )
. m. d. knudson , d. l. hanson , j. e. bailey , c. a. hall , j. r. asay , and c. deeney , phys .
b * 69 * , 144209 ( 2004 ) .
g. w. collins , l. b. da silva , p. celliers , d. m. gold , m. e. foord , r. j. wallace , a. ng , s. v. weber , k. s. budil , and r. cauble , science * 281 * , 1178 ( 1998 ) .
t. r. boehly , d. g. hicks , p. m. celliers , t. j. b. collins , r. earley , j. h. eggert , d. jacobs - perkins , s. j. moon , e. vianello , d. d. meyerhofer , and g. w. collins , phys .
plasmas * 11 * , l49 ( 2004 ) .
d. g. hicks , t. r. boehly , p. m. celliers , j. h. eggert , s. j. moon , d. d. meyerhofer , and g. w. collins , phys .
b * 79 * , 014112 ( 2009 ) .
p. loubeyre , s. btygoo , j. eggert , p. m. celliers , d. k. spaulding , j. r. rygg , t. r. boehly , g. w. collins , and r. jeanloz , phys .
b * 86 * , 144115 ( 2012 ) .
b. holst , r. redmer , and m. p. desjarlais , phys .
b * 77 * , 184201 ( 2008 ) .
f. lambert , ph.d .
thesis , universit paris xi .
commissariat l nergie atomique , 2007 .
b. bernu and p. vieillefosse , phys .
a * 18 * , 2345 ( 1978 ) .
z. donk , b. nyri , l. szalai , and s. holl , phys .
lett . * 81 * , 1622 ( 1998 ) ; z. donk and b. nyri , phys .
plasmas * 7 * , 45 ( 2000 ) .
j. daligault , phys .
lett . * 96 * , 065003 ( 2006 ) ; * 103 * , 029901(e ) ( 2009 ) . j. p. hansen , i. r. mcdonald , and e. l. pollock , phys
a * 11 * , 1025 ( 1975 ) .
r. m. more , adv . at .
* 21 * , 305 ( 1985 ) . | thermophysical properties of hydrogen , helium , and hydrogen - helium mixtures have been investigated in the warm dense matter regime at electron number densities ranging from @xmath0/m@xmath1 and temperatures from 4000 to 20000 k via quantum molecular dynamics simulations .
we focus on the dynamical properties such as the equation of states , diffusion coefficients , and viscosity . mixing rules ( density matching , pressure matching , and binary ionic mixing rules )
have been validated by checking composite properties of pure species against that of the fully interacting mixture derived from qmd simulations .
these mixing rules reproduce pressures within 10% accuracy , while it is 75 % and 50 % for the diffusion and viscosity , respectively .
binary ionic mixing rule moves the results into better agreement .
predictions from one component plasma model are also provided and discussed . | [
2,
109,
692,
113,
109,
10635,
113,
449,
143,
860,
3106,
110,
158,
110,
108,
2725,
1746,
110,
108,
111,
5873,
1696,
113,
13535,
143,
5124,
110,
158,
111,
38585,
143,
178,
110,
158,
365,
3837,
1436,
113,
281,
1367,
111,
1972,
117,
146,
209,
113,
4701,
820,
155,
163,
113,
1442,
2203,
1160,
118,
78716,
2650,
757,
46819,
110,
107,
115,
109,
799,
201,
110,
108,
145,
799,
12584,
10403,
3248,
143,
15593,
208,
252,
110,
158,
18486,
113,
109,
860,
3106,
111,
2725,
1746,
143,
29323,
110,
108,
813,
110,
108,
111,
6606,
31718,
47312,
110,
158,
118,
5124,
233,
178,
44575,
110,
107,
106,
109,
860,
3106,
137,
129,
5215,
190,
228,
972,
120,
117,
110,
108,
4532,
135,
109,
609,
17670,
26834,
3241,
113,
27944,
111,
109,
2606,
1286,
110,
107,
106,
237,
110,
108,
860,
3106,
110,
108,
29323,
110,
108,
111,
31718,
47312,
118,
5124,
233,
178,
44575,
127,
1848,
110,
108,
111,
109,
15593,
208,
252,
602,
127,
1711,
122,
109,
602,
135,
2785,
1581,
111,
15593,
208,
252,
451,
8035,
5873,
1581,
110,
107,
1,
0,
0,
0,
0,
0,
0,
0,
0,
0,
0,
0,
0,
0,
0,
0,
0,
0,
0,
0,
0,
0,
0,
0,
0,
0,
0,
0,
0,
0,
0,
0,
0,
0,
0,
0,
0,
0,
0,
0,
0,
0,
0,
0,
0,
0,
0,
0,
0,
0,
0,
0,
0,
0,
0,
0,
0,
0,
0,
0,
0,
0,
0,
0,
0,
0,
0,
0,
0,
0,
0,
0,
0,
0
] |
groups generated by finite automata ( groups of automata or automaton groups ) were formally introduced at the beginning of 1960 s @xcite , but more substantial work on this remarkable class of groups started only in 1970 s after aleshin @xcite confirmed a conjecture by glushkov @xcite that these groups could be used to study problems of burnside type ( note that groups of automata should not be confused with automatic groups as described in @xcite ) . it was observed in 1960 s and 1970 s that groups of automata are closely related to iterated wreath products ( pioneering work in this direction is due to kaloujnin @xcite ) and that the theory of such groups could be studied by using the language of tables developed by kaloujnin @xcite and sushchanskii @xcite .
even more intensive study of groups of finite automata started in the beginning of 1980 s after the development of some new ideas such as self - similarity , contracting properties , and a geometric realization as groups acting on rooted trees .
these developments allowed for elegant constructions of burnside groups @xcite and pushed the study of groups of automata in many directions : analysis @xcite , geometry @xcite , probability @xcite , dynamics @xcite , formal languages @xcite , etc .
two well known and important problems were solved using groups of automata in the early 1980 s , namely milnor problem @xcite on intermediate growth and day problem @xcite on amenability . a 5-state automaton constructed in @xcite ( on the right in figure [ 3automata ] ) generates a 2-group , denoted @xmath0 .
it was shown in @xcite that @xmath0 has intermediate growth ( between polynomial and exponential ) .
this led to construction of other examples of this type @xcite and also made important contribution to and impact on the theory of invariant means on groups @xcite initiated by von neumann @xcite by providing an example of amenable , but not elementary amenable group ( in the sense of day @xcite ) . among the most interesting newer developments
is the spectral theory of groups generated by finite automata and graphs associated to such groups @xcite .
for instance , automaton groups provided first examples of regular graphs realized as schreier graphs of groups for which the spectrum of the combinatorial laplacian is a cantor set @xcite .
further , the realization of the lamplighter group @xmath1 as automaton group ( bottom left in figure [ 3automata ] ) was crucial in the proof that this group has a pure point spectrum ( with respect to a system of generators related to the states of the automaton ) and thus has discrete spectral measure , which was completely described @xcite .
this , in turn , led to a construction @xcite of a 7-dimensional closed manifold with non - integer third @xmath2-betti number providing a counterexample to the strong atiyah conjecture @xcite .
another fundamental recent discovery is the relation of groups of automata to holomorphic dynamics @xcite .
namely , it is shown that to every rational map @xmath3 on the riemann sphere with finite postcritical set one can associate a finite automaton generating a group , denoted @xmath4 and called iterated monodromy group of @xmath5 . the geometry and the topology of the schreier graphs of @xmath4 is closely related to the geometry of the julia set of @xmath5 .
figure [ b - schreier5 ] depicts a schreier graph associated to an automaton group , denoted @xmath6 and called basilica group .
its reminiscence to the julia set of the map @xmath7 is related to the fact that @xmath6 is the iterated monodromy group @xmath8 of the holomorphic map @xmath9 @xcite ) .
groups of automata represent the basis of the theory of self - similar groups and actions @xcite and are related to the study of belyi polynomials and dessins denfants of grothendieck @xcite .
the use of iterated monodromy groups was crucial in the recent solution of hubbard s twisted rabbit problem in @xcite .
one of the most important developments in the theory of automaton groups is the introduction of groups with branch structure , providing a link to just - infinite groups @xcite and groups of finite width @xcite .
in particular , a problem suggested by zelmanov was solved by using the profinite completion of @xmath0 @xcite .
the problem of gromov on uniformly exponential growth was solved recently by using branch automaton groups @xcite .
an unexpected link of groups of automata and their profinite completions to galois theory was found by r. pink ( private communication ) and aitken , hajir and maire @xcite , while n. boston @xcite related branch groups of automata to fontaine - mazur conjecture and other problems in number theory .
the class of branch groups is also a new source for infinitely presented groups , for which the presentation can be written in a recursive form ( see @xcite ) .
a recent observation @xcite is that automaton groups and their schreier graphs stand behind the famous hanoi towers problem ( see @xcite ) and some of its generalizations @xcite .
there are indications that spectral properties of groups generated by finite automata could be used in the study of kaplansky conjecture on idempotents ( and thus also baum - connes conjecture @xcite and novikov conjecture @xcite ) , dixmier unitarizability problem @xcite , and for construction of new families of expanders , and perhaps even ramanujan , graphs @xcite . in this article
we are going to describe some progress which was achieved during the last few years in the problem of classification of automaton groups
. two important characteristics of an automaton are the cardinalities @xmath10 and @xmath11 of the set of states and the alphabet , respectively , and the pair @xmath12 is a natural measure of complexity of an automaton and of the group it generates .
the groups of complexity @xmath13 are classified @xcite and there are only @xmath14 such groups ( see theorem [ thm_class2states ] in section [ sec_proofs ] here ) .
the problem of classification of @xmath15 groups or @xmath16 groups is much harder .
the current text represents the progress being made by the research group at texas a&m university over the last few years toward classification of @xmath15 groups .
the total number of invertible automata of complexity @xmath15 is @xmath17 .
however , the number of non - isomorphic groups generated by these automata is much smaller .
there are no more than @xmath18 pairwise non - isomorphic groups of complexity @xmath15 .
the proof of this theorem is too long to be presented here ( even the list of all groups takes a lot of space ) . instead
, we have chosen for this article a set of @xmath19 automata generating @xmath20 groups ( among the most interesting in this class , in our opinion ) , which we list in the form of a table .
the table provided here is a part of the table listing the whole set of @xmath18 groups .
we keep the numeration system from the whole table ( the rule for numeration is explained in section [ sec_approach ] ) .
major results obtained for the whole family are the following theorems . the numbers in the brackets indicate the numbers of corresponding automata in the class .
there are @xmath14 finite groups in the class : @xmath21 [ 1 ] , @xmath22 [ 1090 ] , @xmath23 [ 730 ] , @xmath24 [ 847 ] , @xmath25 [ 802 ] and @xmath26 [ 748 ] .
there are @xmath14 abelian groups in the class : @xmath21 [ 1 ] , @xmath22 [ 1090 ] , @xmath23 [ 730 ] , @xmath25 [ 802 ] , @xmath27 [ 731 ] and @xmath28 [ 771 ] .
note that there are also virtually abelian groups in this class ( having @xmath29 , @xmath30 [ 2212 ] , @xmath31 [ 752 ] or @xmath32 [ 968 ] as subgroups of finite index ) .
[ thm : class_free ] the only free non - abelian group in the class is the free group of rank 3 generated by the aleshin - vorobets automaton [ 2240 ] .
moreover , the isomorphism class of this automaton group coincides with its equivalence class under symmetry .
the definition of symmetric automata is given in section [ sec_approach ] .
there are no infinite torsion groups in the class .
we do not provide the complete proofs of these theorems ( by the reason explained above ) .
instead , we give here some information about each of the chosen groups and include proofs of most facts .
properties that are in our focus are the contracting property , self - replication , torsion , relations ( we list the relators up to length @xmath33 ) , rank of quotients of the stabilizers series , shape of the related schreier graphs .
the article is organized as follows .
we start with a general information about rooted trees and their automorphisms .
then we provide quick introduction to the theory of automaton groups .
we continue with the definition of schreier graphs and explain how they naturally appear for the actions on rooted trees . then we list @xmath19 automata generating @xmath20 groups together with some of their properties . in the last section we give proofs of many facts related to the groups in the list .
the last part also contains some more general results ( such as an algorithm detecting transitivity of an element and a criterion for group transitivity on the binary tree ) .
we recommend the articles @xcite and the book @xcite to the reader who is interested in becoming more familiar with automaton groups .
let @xmath34 be fixed and let @xmath35 be the alphabet @xmath36 .
the set of words @xmath37 over @xmath35 ( the free monoid over @xmath35 ) can be given the structure of a _ regular rooted labeled @xmath38-ary tree _
@xmath39 in which the empty word @xmath40 is the _ root _ , the _ level _ @xmath11 in @xmath39 consists of the words of length @xmath11 over @xmath35 and every vertex @xmath41 has @xmath38 children , labeled by @xmath42 , for @xmath43 .
denote by @xmath44 the group of automorphisms of @xmath39 .
let @xmath5 be an automorphism in @xmath44 .
any such automorphism can be decomposed as @xmath45 where @xmath46 , for @xmath43 , are automorphisms of @xmath39 and @xmath47 is a permutation of @xmath35 .
the automorphisms @xmath46 ( also denoted by @xmath48 ) , @xmath43 , are called ( the first level ) _ sections _ of @xmath5 and each one acts as an automorphism on the subtree @xmath49 hanging below the vertex @xmath50 in @xmath39 consisting of the words in @xmath37 that start with @xmath50 ( any such subtree is canonically isomorphic to the whole tree ) .
the action of @xmath5 is decomposed in two steps .
first the @xmath38-tuple @xmath51 acts on the @xmath38 subtrees hanging below the root , and then the permutation @xmath47 , called the _ root permutation _ of @xmath5 , permutes these @xmath38 subtrees .
thus the action of @xmath5 from on @xmath37 is given by @xmath52 for @xmath50 a letter in @xmath35 and @xmath53 a word over @xmath35 .
further iterations of the decomposition ( [ decomposition ] ) yield the second level sections @xmath54 , @xmath55 , and so on .
algebraically , we have @xmath56 where @xmath57 is the _ permutational wreath product _ in which the coordinates of @xmath58 are permuted by @xmath59
. iterations of the decomposition show that @xmath44 has the structure of an iterated wreath product @xmath60 .
thus @xmath44 is a pro - finite group and in particular , all of its subgroups are residually finite . an obvious and natural sequence of normal subgroups of finite index intersecting trivially is the sequence of level stabilizers . the @xmath11-th _ level stabilizer _
@xmath61 of a group @xmath62 consists of those tree automorphisms in @xmath63 that fix the vertices in @xmath39 up to level @xmath11 .
the group @xmath44 is obviously an uncountable object .
we are interested in finitely generated subgroups of @xmath44 that exhibit some important features of @xmath44 .
one such feature is self - similarity .
a group @xmath63 of tree automorphisms is _ self - similar _ if , for every @xmath64 in @xmath63 and a letter @xmath50 in @xmath35 there exists a letter @xmath65 in @xmath35 and an element @xmath66 in @xmath63 such that @xmath67 for all words @xmath53 over @xmath35 . another way to express self - similarity of a group @xmath63 of tree automorphisms is to say that every section @xmath68 of every element @xmath64 in @xmath63 is again an element of @xmath63 .
the full tree automorphism group @xmath44 is clearly self - similar ( see ( [ fxw ] ) ) .
a self - similar group @xmath63 embeds in the permutational wreath product @xmath69 by @xmath70
consider a finite system of recursive relations @xmath71 where each symbol @xmath72 , @xmath73 , @xmath74 , is equal to one of the symbols @xmath75 and @xmath76 .
the system ( [ f^0 ] ) has a unique solution in @xmath44 .
the action of @xmath77 on @xmath39 is given recursively by @xmath78 .
the group generated by the automorphisms @xmath75 is finitely generated self - similar group of automorphisms of @xmath39 .
this group can be described by a finite invertible automaton ( just called automaton in the rest of the article ) .
[ automaton ] a _ finite invertible automaton _ @xmath79 is a @xmath80-tuple @xmath81 where @xmath82 is a finite set of _ states _ , @xmath35 is a finite _ alphabet _ of cardinality @xmath34 , @xmath83 is a map , called _ output map _ , @xmath84 is a map , called _ transition map _ , and for each state @xmath85 in @xmath82 , the restriction @xmath86 given by @xmath87 is a permutation , i.e. @xmath88 . the automaton @xmath81 reads words from @xmath37 and provides output words that are also in @xmath37 . the behavior is encoded in the output and transition maps .
an _ initial automaton _
@xmath89 is just an automaton @xmath79 with a distinguished state @xmath90 selected as an initial state .
we first informally describe the action of the initial automaton @xmath89 on @xmath37 .
the automaton starts at the state @xmath85 , reads the first input letter @xmath91 , outputs the letter @xmath92 and changes its state to @xmath93 .
the rest of the input word is handled by the new state @xmath94 in the same fashion ( in fact it is handled by the initial automaton @xmath95 ) .
formally , the action of the states of the automaton @xmath79 on @xmath37 can be described by extending the output function @xmath96 to a function @xmath97 recursively by @xmath98 for all states @xmath85 in @xmath82 , letters @xmath43 and words @xmath53 over @xmath35 .
then the action of the initial automaton @xmath89 is defined by @xmath99 , for words @xmath100 over @xmath35 .
in fact , ( [ rhoext ] ) shows that each initial automaton @xmath89 , @xmath90 , defines a tree automorphism , denoted by @xmath85 , defined by @xmath101 where the section @xmath102 is the state @xmath103 and the root permutation @xmath104 is the permutation @xmath105 .
given an automaton @xmath81 , the group of tree automorphisms generated by the states of @xmath79 is denoted by @xmath106 and called the _ automaton group _ defined by @xmath79 . the generating set @xmath82 is called the _ standard _ generating set of @xmath106 .
_ boundary _ of the tree @xmath39 , denoted @xmath107 , is the set @xmath108 of words over @xmath35 that are infinite to the right ( infinite geodesic rays in @xmath39 starting at the root ) .
it has a natural metric ( infinite words are close if they agree on long finite prefixes ) and the group of isometries @xmath109 is canonically isomorphic to @xmath44 .
thus the action of the automaton group @xmath106 on @xmath39 can be extended to an isometric action on @xmath107 .
in fact , and are valid for infinite words @xmath53 as well .
an automaton @xmath79 can be represented by a labeled directed graph , called moore diagram , in which the vertices are the states of the automaton , each state @xmath85 is labeled by its own root permutation @xmath104 and , for each pair @xmath110 , there is an edge from @xmath85 to @xmath111 labeled by @xmath50 .
for example , the 5-state automaton in the right half of figure [ 3automata ] generates the group @xmath0 mentioned in the introduction ( @xmath112 denotes the permutation exchanging @xmath113 and @xmath114 ) .
the two 2-state automata given on the left of figure [ 3automata ] are the so called _ adding machine _ ( top ) , which generates the infinite cyclic group @xmath29 and the _ lamplighter automaton _ ( bottom ) generating @xmath115 .
recursion relations of type ( [ f^0 ] ) for the adding machine and the lamplighter automaton are given by @xmath116 respectively .
various classes of automaton groups deserve special attention .
an automaton group @xmath117 is _ contracting _ if there exist constants @xmath118 , @xmath119 , and @xmath120 , with @xmath121 , such that @xmath122 , for all vertices @xmath41 of length at least @xmath120 and @xmath123 ( the length is measured with respect to the standard generating set @xmath82 )
. for sufficiently long elements @xmath64 this means that the length of its sections at vertices on levels deeper than @xmath120 is strictly shorter than the length of @xmath64 .
this length shortening leads to an equivalent definition of a contracting group .
namely , a group @xmath63 of tree automorphisms is contracting if there exists a finite set @xmath124 , such that for every @xmath125 , there exists @xmath126 , such that @xmath127 for all vertices @xmath128 of length not shorter than @xmath120 . the minimal set @xmath129 with this property
is called the _ nucleus _ of @xmath63 .
the contraction property is a key feature of various inductive arguments and algorithms involving the decomposition @xmath130 .
another important class is the class of automaton groups of _ branch type_. branch groups arise as one of the three @xcite possible types of just - infinite groups ( infinite groups for which all proper homomorphic images are finite ) .
every infinite , finitely generated group has a just - infinite image . thus if a class of groups @xmath131 is closed under homomorphic images and if it contains infinite , finitely generated examples then it contains just - infinite examples .
such examples are , in a sense , minimal infinite examples in @xmath131 .
for example , @xmath0 is a branch automaton group that is a just - infinite 2-group .
i.e. , it is an infinite , finitely generated , torsion group that has no proper infinite quotients .
also , the hanoi towers group @xcite and the iterated monodromy group @xmath132 @xcite are branch groups , while @xmath133 is not a branch group , but only weakly branch ( for definitions see @xcite ) . the class of _ polynomially growing automata _ was introduced by sidki in @xcite , where it is proved that no group @xmath106 defined by such an automaton contains free subgroups of rank 2 .
moreover , for a subclass of so called _ bounded automata _ it is known that the corresponding groups are amenable @xcite ( this class of automata , for instance , includes the automata generating @xmath0 , @xmath6 and hanoi towers group on @xmath134 pegs , but not for more pegs ) .
finally , self - replicating groups play an important role .
a self - similar group @xmath63 is called _ self - replicating _ if , for every vertex @xmath100 , the homomorphism @xmath135 from the stabilizer @xmath100 in @xmath63 to @xmath63 , given by @xmath136 , is surjective .
this condition is usually easy to check and , together with transitivity of the action on level 1 , it implies transitivity of the action on all levels . another way to show that a group of automorphisms of the binary tree is level transitive is to use proposition [ prop : transitivity ] .
let us fix some self - similar contracting group acting on @xmath137 by automorphisms . denote by @xmath138 the space of left infinite sequences over @xmath35 .
two elements @xmath139 are said to be _ asymptotically equivalent _ with respect to the action of the group @xmath63 , if there exist a finite set @xmath140 and a sequence @xmath141 of elements in @xmath142 such that @xmath143 for every @xmath144 .
the asymptotic equivalence is an equivalence relation .
moreover , sequences @xmath145 are asymptotically equivalent if and only if there exists a sequence @xmath146 of the elements in the nucleus of @xmath63 such that @xmath147 and @xmath148 , for all @xmath144 .
the quotient space @xmath149 of the topological space @xmath138 by the asymptotic equivalence relation is called the _
limit space _ of the self - similar action of @xmath63 .
the limit space @xmath149 is metrizable and finite - dimensional .
if the group @xmath63 is finitely - generated and level - transitive , then the limit space @xmath149 is connected .
the last decade witnessed a shift in the attention payed to the study of schreier graphs .
let @xmath63 be a group generated by a finite set @xmath150 and let @xmath63 act on a set @xmath151 .
the _ schreier graph _ of the action @xmath152 is the graph @xmath153 with set of vertices @xmath151 and set of edges @xmath154 , where the arrow @xmath155 starts in @xmath65 and ends in @xmath156 . if @xmath157 then the schreier graph @xmath158 of the action of @xmath63 on the @xmath63-orbit of @xmath65 is called _
orbital schreier graph_. let @xmath63 be a subgroup of @xmath44 generated by a finite set @xmath150 ( not necessary self - similar ) . the levels @xmath159 , @xmath160 , are invariant under the action of @xmath63 and we can consider the schreier graphs @xmath161 .
let @xmath162 .
then the pointed schreier graphs @xmath163 converge in the local topology ( topology defined in @xcite ) to the pointed orbital schreier graph @xmath164 .
the limit space of a finitely generated contracting self - similar group @xmath63 can be viewed as a hyperbolic boundary in the following way . for
any given finite generating system @xmath150 of @xmath63 define the self - similarity graph @xmath165 as the graph with set of vertices @xmath137 in which two vertices @xmath166 are connected by an edge if and only if either @xmath167 , for some @xmath168 ( vertical edges ) , or @xmath169 for some @xmath170 ( horizontal edges ) . if the group is contracting then the self - similarity graph @xmath165 is gromov - hyperbolic and its hyperbolic boundary is homeomorphic to the limit space @xmath149 .
the set of horizontal edges of @xmath165 spans the disjoint union of all schreier graphs @xmath171.thus , the schreier graphs @xmath171 in some sense approximate the limit space @xmath149 of the group @xmath63 .
moreover , for many examples of self - similar contracting groups there exists a sequence of numbers @xmath172 such that the metric spaces @xmath173 , where @xmath38 is the combinatorial metric on the graph , converge in the gromov - hausdorff metric to the limit space of the group .
we recall the definition and basic properties of iterated monodromy groups ( img ) .
let @xmath174 be a path connected and locally path connected topological space and let @xmath175 be its open path connected subset .
let @xmath176 be a @xmath38-fold covering .
by @xmath177 we denote the @xmath11-th iteration of the map @xmath5 .
the map @xmath178 , where @xmath179 , is a @xmath180-fold covering . choose an arbitrary base point @xmath181 .
let @xmath182 be the disjoint union of the sets @xmath183 ( these sets are note necessarily disjoint by themselves ) .
the set of pre - images @xmath182 has a natural structure of a rooted @xmath38-ary tree with root @xmath184 in which every vertex @xmath185 is connected to the vertex @xmath186 , @xmath187 .
the fundamental group @xmath188 acts naturally on every set @xmath189 and , in fact , acts by automorphisms on @xmath182 .
_ iterated monodromy group _ @xmath4 of the covering @xmath5 is the quotient of the fundamental group @xmath188 by the kernel of its action on @xmath182 .
it is proved in @xcite that all iterated monodromy groups are self - similar .
this fact provides a connection between holomorphic dynamics and groups generated by automata .
the iterated monodromy group of a sub - hyperbolic rational function is contracting and its limit space is homeomorphic to the julia set of the rational function . in particular , the sequence of schreier graphs @xmath190 of the iterated monodromy group of a sub - hyperbolic rational function can be drawn on the riemann sphere in such a way that they converge in the hausdorff metric to the julia set of the function .
schreier graphs also play a role in computing the spectrum of the markov operator @xmath191 on the group .
namely , given a group @xmath63 generated by a finite set @xmath192 , acting on a tree @xmath37 there is a natural unitary representation of @xmath63 in the space of bounded linear operators @xmath193 given by @xmath194 . the markov operator @xmath195 corresponding to this unitary representation plays an important role .
the spectrum of @xmath191 for a self - similar group @xmath63 is approximated by the spectra of finite dimensional operators arising from the action of @xmath63 on the levels of the tree @xmath37 . for more on this see @xcite .
let @xmath196 be a subspace of @xmath197 spanned by the @xmath198 characteristic functions @xmath199 , of the cylindrical sets corresponding to the @xmath198 vertices on level @xmath11 .
then @xmath196 is invariant under the action of @xmath63 and @xmath200 .
denote by @xmath201 the restriction of @xmath202 on @xmath196 .
then @xmath203 are finite dimensional operators , whose spectra converge to the spectrum of @xmath191 in the sense @xmath204 if @xmath205 is the stabilizer of an infinite word from @xmath108 , then one can consider the markov operator @xmath206 on the schreier graph of @xmath63 with respect to @xmath205
. the following fact is observed in @xcite and can be applied to compute the spectrum of markov operator on the cayley graph of a group in case if @xmath205 is small . if @xmath63 is amenable or the schreier graph @xmath207 ( the schreier graph of the action of @xmath63 on the cosets of @xmath205 ) is amenable then @xmath208 .
the next three sections are devoted to the groups generated by 3-state automata over the 2-letter alphabet @xmath209 .
fix @xmath210 as the set of states .
every @xmath15 automaton is given by @xmath211 where @xmath212 , for @xmath213 , @xmath214 , @xmath215 , and @xmath216 .
a number is assigned to the automaton above by the following formula @xmath217 thus every @xmath15 automaton obtains a unique number in the range from @xmath114 to @xmath218 .
the numbering of the automata is induced by the lexicographic ordering of all automata in the class .
the automata numbered @xmath114 through @xmath219 act trivially on the tree and generate the trivial group .
the automata numbered @xmath220 through @xmath218 generate the group @xmath22 of order @xmath221 , because every element in any of these groups is either trivial , or changes all letters in any word over @xmath35 .
therefore the `` interesting '' automata have numbers @xmath222 through @xmath223 .
denote by @xmath224 the automaton numbered by @xmath11 and by @xmath225 the corresponding group of tree automorphisms .
sometimes , when the context is clear , we use just the number to refer to the corresponding automaton or group . the following operations on automata change neither the group generated by this automaton , nor , essentially , the action of the group on the tree . 1 . passing to inverses of all generators 2 .
permuting the states of the automaton 3 . permuting the letters of the alphabet two automata @xmath226 and @xmath227 that can be obtained from one another using a composition of the operations ( @xmath228)(@xmath229 ) ,
are called _ symmetric_.
if the minimization of an automaton @xmath226 is symmetric to the minimization of an automaton @xmath227 , we say that the automata @xmath226 and @xmath227 are _ minimally symmetric _ and write @xmath230 .
another equivalence relation we consider is the isomorphism of the groups generated by the automata .
the minimal symmetry relation is a refinement of the isomorphism relation , since the same abstract group may have different actions on the binary tree .
there are @xmath231 classes of automata , which are pairwise not minimally symmetric , @xmath33 of which are minimally symmetric to automata with fewer than @xmath134 states .
these @xmath33 classes of automata are subject of theorem [ thm_class2states ] , which states that they generate @xmath14 different groups . at present , it is known that there are at most @xmath18 non - isomorphic groups in the considered class .
in this section we provide information about selected groups in the class of all groups generated by @xmath15 automata .
the groups are selected in such a way that the corresponding proofs in section [ sec_proofs ] show most of the main methods and ideas that were used for the whole class .
the following notation is used : * rels - this is a list of some relators in the group .
all independent relators up to length @xmath20 are included . on some situations additional longer relators
are included . for @xmath232 and @xmath233
there are no relators of length up to @xmath20 and the relators provided in the table are not necessarily the shortest . in many cases ,
the given relations are not sufficient ( for example , some of the groups are not finitely presented ) . *
sf - these numbers represent the size of the factors @xmath234 , for @xmath235 .
* gr - these numbers represent the values of the growth function @xmath236 , for @xmath235 , and generating system @xmath237 , @xmath238 , @xmath239 .
finally , for each automaton in the list a histogram for the spectral density of the operator @xmath240 acting on level @xmath241 of the tree is shown . in some cases ,
in order to show the main ways to prove the group isomorphism , we provide several different automata generating the same group .
@p172ptp174pt@ * automaton number @xmath242 * @p48ptp200pt @xmath243 @xmath244 @xmath245 & group : @xmath246 contracting : _ yes _
self - replicating : _ no _
+ rels : @xmath247 , @xmath248 , @xmath249 , @xmath250 , @xmath251 , @xmath252 , + @xmath253 , @xmath254 , @xmath255 , + @xmath256 , @xmath257 , @xmath258 , + @xmath259 , @xmath260 , + @xmath261 , @xmath262 , + @xmath263 + sf : @xmath264,@xmath265,@xmath266,@xmath267,@xmath268,@xmath269,@xmath270,@xmath271,@xmath272 + gr : 1,4,9,17,30,47,68,93,122,155,192 + & ( 1450,1090)(0,130 ) ( 200,200 ) ( 1200,200)(700,1070 ) ( 45,280)@xmath237 ( 1280,280)@xmath238 ( 820,1035)@xmath239 ( 164,165)@xmath112 ( 1164,152)@xmath114 ( 664,1022)@xmath114 ( 100,100 ) ( 46,216)(100,200)(55,167 ) ( 287,150)(700,0)(1113,150 ) ( 325,109)(287,150)(343,156 ) ( 1300,100 ) ( 1345,167)(1300,200)(1354,216 ) ( 650,983)(250,287 ) ( 297,318)(250,287)(253,343 ) ( 190,10)@xmath273 ( 1155,10)@xmath274 ( 680,77)@xmath275 ( 455,585)@xmath273 + @p172ptp174pt@ * automaton number @xmath276 * @p48ptp200pt @xmath277 @xmath244 @xmath245 & group : contracting : _ no _ self - replicating : _ yes _ + rels :
@xmath278 , @xmath279 , @xmath280 , @xmath281 + sf : @xmath264,@xmath265,@xmath266,@xmath267,@xmath270,@xmath282,@xmath283,@xmath284,@xmath285 + gr : 1,7,37,187,937,4687 + & ( 1450,1090)(0,130 ) ( 200,200 ) ( 1200,200)(700,1070 ) ( 45,280)@xmath237 ( 1280,280)@xmath238 ( 820,1035)@xmath239 ( 164,165)@xmath112 ( 1164,152)@xmath114 ( 664,1022)@xmath114 ( 300,200)(1,0)800 ( 1050,225)(1100,200)(1050,175 ) ( 200,300)(277,733)(613,1020 ) ( 559,1007)(613,1020)(591,969 ) ( 287,150)(700,0)(1113,150 ) ( 325,109)(287,150)(343,156 ) ( 1300,100 ) ( 1345,167)(1300,200)(1354,216 ) ( 650,983)(250,287 ) ( 297,318)(250,287)(253,343 ) ( 230,700)@xmath274 ( 680,240)@xmath275 ( 1155,10)@xmath274 ( 680,77)@xmath275 ( 455,585)@xmath273 + @p172ptp174pt@ * automaton number @xmath286 * @p48ptp200pt @xmath243 @xmath287 @xmath245 & group : @xmath26 contracting : _ yes _
self - replicating : _ no _
+ rels : @xmath247 , @xmath248 , @xmath249 , @xmath250 , @xmath288 , @xmath289 + sf : @xmath264,@xmath265,@xmath266,@xmath290,@xmath290,@xmath290,@xmath290,@xmath290,@xmath290 + gr : 1,4,8,12,15,16,16,16,16,16,16 + & ( 1450,1090)(0,130 ) ( 200,200 ) ( 1200,200)(700,1070 ) ( 45,280)@xmath237 ( 1280,280)@xmath238 ( 820,1035)@xmath239 ( 164,165)@xmath112 ( 1164,152)@xmath114 ( 664,1022)@xmath114 ( 100,100 ) ( 46,216)(100,200)(55,167 ) ( 287,150)(700,0)(1113,150 ) ( 325,109)(287,150)(343,156 ) ( 750,983)(1150,287 ) ( 753,927)(750,983)(797,952 ) ( 650,983)(250,287 ) ( 297,318)(250,287)(253,343 ) ( 190,10)@xmath273 ( 890,585)@xmath274 ( 680,77)@xmath275 ( 455,585)@xmath273 + @p172ptp174pt@ * automaton number @xmath291 * @p48ptp200pt @xmath277 @xmath287 @xmath245 & group : contracting : _ no _ self - replicating : _ yes _
+ rels : @xmath292 + @xmath293 , @xmath294 + @xmath295 , @xmath296 + @xmath297 + sf : @xmath264,@xmath265,@xmath266,@xmath267,@xmath270,@xmath282,@xmath283,@xmath284,@xmath285 + gr : 1,7,37,187,937,4687 + & ( 1450,1090)(0,130 ) ( 200,200 ) ( 1200,200)(700,1070 ) ( 45,280)@xmath237 ( 1280,280)@xmath238 ( 820,1035)@xmath239 ( 164,165)@xmath112 ( 1164,152)@xmath114 ( 664,1022)@xmath114 ( 300,200)(1,0)800 ( 1050,225)(1100,200)(1050,175 ) ( 200,300)(277,733)(613,1020 ) ( 559,1007)(613,1020)(591,969 ) ( 287,150)(700,0)(1113,150 ) ( 325,109)(287,150)(343,156 ) ( 750,983)(1150,287 ) ( 753,927)(750,983)(797,952 ) ( 650,983)(250,287 ) ( 297,318)(250,287)(253,343 ) ( 230,700)@xmath274 ( 680,240)@xmath275 ( 890,585)@xmath274 ( 680,77)@xmath275 ( 455,585)@xmath273 + @p172ptp174pt@ * automaton number @xmath298 * @p48ptp200pt @xmath277 @xmath299 @xmath245 & group : @xmath28 contracting : _ yes _
self - replicating : _ yes _ + rels : @xmath238 , @xmath300 + sf : @xmath264,@xmath265,@xmath301,@xmath266,@xmath290,@xmath302,@xmath267,@xmath303,@xmath268 + gr : 1,5,13,25,41,61,85,113,145,181,221 + limit space : @xmath221-dimensional torus @xmath304 & ( 1450,1090)(0,130 ) ( 200,200 ) ( 1200,200)(700,1070 ) ( 45,280)@xmath237 ( 1280,280)@xmath238 ( 820,1035)@xmath239 ( 164,165)@xmath112 ( 1164,152)@xmath114 ( 664,1022)@xmath114 ( 300,200)(1,0)800 ( 1050,225)(1100,200)(1050,175 ) ( 200,300)(277,733)(613,1020 ) ( 559,1007)(613,1020)(591,969 ) ( 1300,100 ) ( 1345,167)(1300,200)(1354,216 ) ( 650,983)(250,287 ) ( 297,318)(250,287)(253,343 ) ( 230,700)@xmath274 ( 680,240)@xmath275 ( 1080,10)@xmath273 ( 455,585)@xmath273 + @p172ptp174pt@ * automata number @xmath305 and @xmath306 * @p48ptp12ptp48ptp200pt @xmath243 @xmath307 @xmath245 & 783 : & @xmath308 @xmath307 @xmath245&group : @xmath309 contracting : _ yes _ self - replicating : _ yes _ + rels :
@xmath247 , @xmath248 , @xmath249 , @xmath250 , @xmath310 , + @xmath311 , @xmath312 , + @xmath313 , @xmath314 + sf : @xmath264,@xmath265,@xmath301,@xmath290,@xmath267,@xmath315,@xmath316,@xmath317,@xmath318 + gr : 1,4,9,17,30,51,85,140,229,367,579 + limit space : & ( 1450,1090)(0,130 ) ( 200,200 ) ( 1200,200)(700,1070 ) ( 45,280)@xmath237 ( 1280,280)@xmath238 ( 820,1035)@xmath239 ( 164,165)@xmath112 ( 1164,152)@xmath114 ( 664,1022)@xmath114 ( 100,100 ) ( 46,216)(100,200)(55,167 ) ( 1300,100 ) ( 1345,167)(1300,200)(1354,216 ) ( 750,983)(1150,287 ) ( 753,927)(750,983)(797,952 ) ( 650,983)(250,287 ) ( 297,318)(250,287)(253,343 ) ( 190,10)@xmath273 ( 890,585)@xmath274 ( 1160,10)@xmath275 ( 455,585)@xmath273 + @p172ptp174pt@ * automaton number @xmath319 * @p48ptp200pt @xmath320 @xmath321 @xmath245 & group : @xmath28 contracting : _ yes _
self - replicating : _ yes _ + rels : @xmath322 , @xmath300 + sf : @xmath264,@xmath265,@xmath301,@xmath266,@xmath290,@xmath302,@xmath267,@xmath303,@xmath268 + gr : 1,7,21,43,73,111,157,211,273,343,421 + limit space : @xmath221-dimensional torus @xmath304 & ( 1450,1090)(0,130 ) ( 200,200 ) ( 1200,200)(700,1070 ) ( 45,280)@xmath237 ( 1280,280)@xmath238 ( 820,1035)@xmath239 ( 164,165)@xmath112 ( 1164,152)@xmath114 ( 664,1022)@xmath114 ( 100,100 ) ( 46,216)(100,200)(55,167 ) ( 300,200)(1,0)800 ( 1050,225)(1100,200)(1050,175 ) ( 750,983)(1150,287 ) ( 753,927)(750,983)(797,952 ) ( 650,983)(250,287 ) ( 297,318)(250,287)(253,343 ) ( 680,240)@xmath274 ( 193,10)@xmath275 ( 820,585)@xmath273 ( 455,585)@xmath273 + @p172ptp174pt@ * automaton number @xmath323 * @p48ptp200pt @xmath308 @xmath324 @xmath325 & group : @xmath326 contracting : _ no _ self - replicating : _ no _
+ rels : @xmath247 , @xmath248 , @xmath249 + sf : @xmath264,@xmath265,@xmath266,@xmath302,@xmath303,@xmath269,@xmath327,@xmath272,@xmath328 + gr : 1,4,10,22,46,94,190,382,766,1534 + & ( 1450,1090)(0,130 ) ( 200,200 ) ( 1200,200)(700,1070 ) ( 45,280)@xmath237 ( 1280,280)@xmath238 ( 820,1035)@xmath239 ( 164,165)@xmath112 ( 1164,152)@xmath114 ( 664,1022)@xmath114 ( 200,300)(277,733)(613,1020 ) ( 559,1007)(613,1020)(591,969 ) ( 287,150)(700,0)(1113,150 ) ( 325,109)(287,150)(343,156 ) ( 1300,100 ) ( 1345,167)(1300,200)(1354,216 ) ( 650,983)(250,287 ) ( 297,318)(250,287)(253,343 ) ( 1200,300)(1123,733)(787,1020 ) ( 1216,354)(1200,300)(1167,345 ) ( 150,700)@xmath273 ( 680,77)@xmath274 ( 1160,10)@xmath275 ( 1115,700)@xmath274(460,585)@xmath275 + @p172ptp174pt@ * automaton number @xmath329 * @p48ptp200pt @xmath277 @xmath299 @xmath325 & group : _ basilica group _
contracting : _ yes _ self - replicating : _ yes _ + rels : @xmath238 , @xmath330 , @xmath331 , + @xmath332 , @xmath333 , + @xmath334 , @xmath335 , + @xmath336 + sf : @xmath264,@xmath265,@xmath266,@xmath267,@xmath270,@xmath282,@xmath283,@xmath284,@xmath285 + gr : 1,5,17,53,153,421,1125,2945,7545 + limit space : & ( 1450,1090)(0,130 ) ( 200,200 ) ( 1200,200)(700,1070 ) ( 45,280)@xmath237 ( 1280,280)@xmath238 ( 820,1035)@xmath239 ( 164,165)@xmath112 ( 1164,152)@xmath114 ( 664,1022)@xmath114 ( 300,200)(1,0)800 ( 1050,225)(1100,200)(1050,175 ) ( 200,300)(277,733)(613,1020 ) ( 559,1007)(613,1020)(591,969 ) ( 1300,100 ) ( 1345,167)(1300,200)(1354,216 ) ( 650,983)(250,287 ) ( 297,318)(250,287)(253,343 ) ( 1200,300)(1123,733)(787,1020 ) ( 1216,354)(1200,300)(1167,345 ) ( 230,700)@xmath274 ( 680,240)@xmath275 ( 1080,10)@xmath273 ( 1115,700)@xmath274(460,585)@xmath275 + @p172ptp174pt@ * automaton number @xmath337 * @p48ptp200pt @xmath320 @xmath307 @xmath325 & group : contracting : _ no _ self - replicating : _ yes _ + rels :
@xmath338 , @xmath339 , + @xmath340 , @xmath341 , @xmath342 , @xmath343 , @xmath344 , + @xmath345 , @xmath346 + sf : @xmath264,@xmath265,@xmath266,@xmath303,@xmath327,@xmath347,@xmath348,@xmath349,@xmath350 + gr : 1,7,35,165,758,3460 + & ( 1450,1090)(0,130 ) ( 200,200 ) ( 1200,200)(700,1070 ) ( 45,280)@xmath237 ( 1280,280)@xmath238 ( 820,1035)@xmath239 ( 164,165)@xmath112 ( 1164,152)@xmath114 ( 664,1022)@xmath114 ( 100,100 ) ( 46,216)(100,200)(55,167 ) ( 300,200)(1,0)800 ( 1050,225)(1100,200)(1050,175 ) ( 1300,100 ) ( 1345,167)(1300,200)(1354,216 ) ( 750,983)(1150,287 ) ( 753,927)(750,983)(797,952 ) ( 650,983)(250,287 ) ( 297,318)(250,287)(253,343 ) ( 1200,300)(1123,733)(787,1020 ) ( 1216,354)(1200,300)(1167,345 ) ( 680,240)@xmath274 ( 193,10)@xmath275 ( 890,585)@xmath274 ( 1160,10)@xmath275 ( 1115,700)@xmath274(460,585)@xmath275 + @p172ptp174pt@ * automaton number @xmath351 * @p48ptp200pt @xmath352 @xmath307 @xmath325 & group : contracting : _ no _ self - replicating : _ yes _ + rels : @xmath353 + @xmath354 , @xmath355 + @xmath356 + sf : @xmath264,@xmath265,@xmath266,@xmath303,@xmath327,@xmath357,@xmath358,@xmath349,@xmath350 + gr : 1,7,37,187,937,4687 + & ( 1450,1090)(0,130 ) ( 200,200 ) ( 1200,200)(700,1070 ) ( 45,280)@xmath237 ( 1280,280)@xmath238 ( 820,1035)@xmath239 ( 164,165)@xmath112 ( 1164,152)@xmath114 ( 664,1022)@xmath114 ( 100,100 ) ( 46,216)(100,200)(55,167 ) ( 200,300)(277,733)(613,1020 ) ( 559,1007)(613,1020)(591,969 ) ( 1300,100 ) ( 1345,167)(1300,200)(1354,216 ) ( 750,983)(1150,287 ) ( 753,927)(750,983)(797,952 ) ( 650,983)(250,287 ) ( 297,318)(250,287)(253,343 ) ( 1200,300)(1123,733)(787,1020 ) ( 1216,354)(1200,300)(1167,345 ) ( 230,700)@xmath274 ( 193,10)@xmath275 ( 890,585)@xmath274 ( 1160,10)@xmath275 ( 1115,700)@xmath274(460,585)@xmath275 + @p172ptp174pt@ * automaton number @xmath359 * @p48ptp200pt @xmath277 @xmath360 @xmath325 & group : _ baumslag - solitar group @xmath361 _ contracting : _ no _ self - replicating : _ yes _ + rels : @xmath362 , @xmath363 + sf : @xmath264,@xmath265,@xmath266,@xmath290,@xmath267,@xmath268,@xmath269,@xmath270,@xmath271 + gr : 1,7,33,127,433,1415 + & ( 1450,1090)(0,130 ) ( 200,200 ) ( 1200,200)(700,1070 ) ( 45,280)@xmath237 ( 1280,280)@xmath238 ( 820,1035)@xmath239 ( 164,165)@xmath112 ( 1164,152)@xmath114 ( 664,1022)@xmath114 ( 300,200)(1,0)800 ( 1050,225)(1100,200)(1050,175 ) ( 200,300)(277,733)(613,1020 ) ( 559,1007)(613,1020)(591,969 ) ( 287,150)(700,0)(1113,150 ) ( 325,109)(287,150)(343,156 ) ( 750,983)(1150,287 ) ( 753,927)(750,983)(797,952 ) ( 650,983)(250,287 ) ( 297,318)(250,287)(253,343 ) ( 1200,300)(1123,733)(787,1020 ) ( 1216,354)(1200,300)(1167,345 ) ( 230,700)@xmath274 ( 680,240)@xmath275 ( 680,77)@xmath274 ( 890,585)@xmath275 ( 1115,700)@xmath274(460,585)@xmath275 + @p172ptp174pt@ * automaton number @xmath364 * @p48ptp200pt @xmath365 @xmath366 @xmath325 & group : @xmath367 contracting : _ yes _
self - replicating : _ yes _
+ rels : @xmath247 , @xmath248 , @xmath249 , @xmath368 , @xmath369 + sf : @xmath264,@xmath265,@xmath266,@xmath303,@xmath327,@xmath357,@xmath358,@xmath370,@xmath371 + gr : 1,4,10,22,46,94,184,352,664,1244,2296,4198,7612 + limit space : & ( 1450,1090)(0,130 ) ( 200,200 ) ( 1200,200)(700,1070 ) ( 45,280)@xmath237 ( 1280,280)@xmath238 ( 820,1035)@xmath239 ( 164,165)@xmath112 ( 1164,152)@xmath114 ( 664,1022)@xmath114 ( 300,200)(1,0)800 ( 1050,225)(1100,200)(1050,175 ) ( 1300,100 ) ( 1345,167)(1300,200)(1354,216 ) ( 750,983)(1150,287 ) ( 753,927)(750,983)(797,952 ) ( 650,983)(250,287 ) ( 297,318)(250,287)(253,343 ) ( 1200,300)(1123,733)(787,1020 ) ( 1216,354)(1200,300)(1167,345 ) ( 650,250)@xmath273 ( 1155,10)@xmath274 ( 890,585)@xmath275 ( 1115,700)@xmath274(460,585)@xmath275 + @p172ptp174pt@ * automaton number @xmath372 * @p48ptp200pt @xmath320 @xmath299 @xmath373 & group : contracting : _ no _ self - replicating : _ yes _ + rels : @xmath238 , @xmath374 + sf : @xmath264,@xmath265,@xmath266,@xmath267,@xmath270,@xmath282,@xmath283,@xmath284,@xmath285 + gr : 1,5,17,53,161,475,1387 + & ( 1450,1090)(0,130 ) ( 200,200 ) ( 1200,200)(700,1070 ) ( 45,280)@xmath237 ( 1280,280)@xmath238 ( 820,1035)@xmath239 ( 164,165)@xmath112 ( 1164,152)@xmath114 ( 664,1022)@xmath114 ( 100,100 ) ( 46,216)(100,200)(55,167 ) ( 300,200)(1,0)800 ( 1050,225)(1100,200)(1050,175 ) ( 1300,100 ) ( 1345,167)(1300,200)(1354,216 ) ( 650,983)(250,287 ) ( 297,318)(250,287)(253,343 ) ( 700,1211 ) ( 820,1168)(771,1141)(779,1196 ) ( 680,240)@xmath274 ( 193,10)@xmath275 ( 1080,10)@xmath273 ( 545,1261)@xmath274 ( 460,585)@xmath275 + @p172ptp174pt@ * automaton number @xmath375 * @p48ptp200pt @xmath277 @xmath307 @xmath373 & group : contains the lamplighter group contracting : _ no _ self - replicating : _ yes _ + rels : @xmath376 , @xmath377 , @xmath378 , + @xmath379 , @xmath380 , @xmath381 , + @xmath382 , @xmath383 , @xmath384 , + @xmath385 , @xmath386 , + @xmath387 , @xmath388 , + @xmath389 , @xmath390 , + @xmath391 , @xmath392 , + @xmath393 , + @xmath394 , + @xmath395 + sf : @xmath264,@xmath265,@xmath266,@xmath303,@xmath327,@xmath347,@xmath348,@xmath349,@xmath350 + gr : 1,7,33,143,597,2465 + & ( 1450,1090)(0,130 ) ( 200,200 ) ( 1200,200)(700,1070 ) ( 45,280)@xmath237 ( 1280,280)@xmath238 ( 820,1035)@xmath239 ( 164,165)@xmath112 ( 1164,152)@xmath114 ( 664,1022)@xmath114 ( 300,200)(1,0)800 ( 1050,225)(1100,200)(1050,175 ) ( 200,300)(277,733)(613,1020 ) ( 559,1007)(613,1020)(591,969 ) ( 1300,100 ) ( 1345,167)(1300,200)(1354,216 ) ( 750,983)(1150,287 ) ( 753,927)(750,983)(797,952 ) ( 650,983)(250,287 ) ( 297,318)(250,287)(253,343 ) ( 700,1211 ) ( 820,1168)(771,1141)(779,1196 ) ( 230,700)@xmath274 ( 680,240)@xmath275 ( 890,585)@xmath274 ( 1160,10)@xmath275 ( 545,1261)@xmath274 ( 460,585)@xmath275 + @p172ptp174pt@ * automaton number @xmath396 * @p48ptp200pt @xmath365 @xmath321 @xmath373 & group : contains @xmath32 as a subgroup of index @xmath397 contracting : _ yes _
self - replicating : _ no _
+ rels : @xmath247 , @xmath248 , @xmath249 , @xmath368 , @xmath256 , + @xmath398 , @xmath369 , @xmath399 , + @xmath400 , @xmath401 , + @xmath311 , @xmath402 + sf : @xmath264,@xmath265,@xmath266,@xmath267,@xmath315,@xmath327,@xmath403,@xmath404,@xmath347 + gr : 1,4,10,22,46,94,184,338,600,1022 + & ( 1450,1090)(0,130 ) ( 200,200 ) ( 1200,200)(700,1070 ) ( 45,280)@xmath237 ( 1280,280)@xmath238 ( 820,1035)@xmath239 ( 164,165)@xmath112 ( 1164,152)@xmath114 ( 664,1022)@xmath114 ( 300,200)(1,0)800 ( 1050,225)(1100,200)(1050,175 ) ( 750,983)(1150,287 ) ( 753,927)(750,983)(797,952 ) ( 650,983)(250,287 ) ( 297,318)(250,287)(253,343 ) ( 700,1211 ) ( 820,1168)(771,1141)(779,1196 ) ( 650,250)@xmath273 ( 820,585)@xmath273 ( 545,1261)@xmath274 ( 460,585)@xmath275 + @p172ptp174pt@ * automaton number @xmath405 * @p48ptp200pt @xmath308 @xmath406 @xmath245 & group : @xmath309 contracting : _ yes _
self - replicating : _ yes _
+ rels : @xmath247 , @xmath249 , @xmath250 , @xmath407 , @xmath408 , + @xmath409 , @xmath410 , @xmath411 , @xmath412 , + @xmath413 , @xmath414 , @xmath415 , @xmath416 , + @xmath417 , @xmath418 + sf : @xmath264,@xmath265,@xmath301,@xmath290,@xmath267,@xmath315,@xmath316,@xmath317,@xmath318,@xmath419 + gr : 1,5,16,40,88,184,376,746,1458 + limit space : & ( 1450,1090)(0,130 ) ( 200,200 ) ( 1200,200)(700,1070 ) ( 45,280)@xmath237 ( 1280,280)@xmath238 ( 820,1035)@xmath239 ( 164,165)@xmath112 ( 1164,165)@xmath112 ( 664,1022)@xmath114 ( 200,300)(277,733)(613,1020 ) ( 559,1007)(613,1020)(591,969 ) ( 287,150)(700,0)(1113,150 ) ( 325,109)(287,150)(343,156 ) ( 1300,100 ) ( 1345,167)(1300,200)(1354,216 ) ( 650,983)(250,287 ) ( 297,318)(250,287)(253,343 ) ( 150,700)@xmath273 ( 1155,10)@xmath274 ( 680,77)@xmath275 ( 455,585)@xmath273 + @p172ptp174pt@ * automaton number @xmath420 * @p48ptp200pt @xmath421 @xmath422 @xmath245 & group : _ klein bottle group _ , virtually @xmath30 contracting : _ yes _ self - replicating : _ no _
+ rels : @xmath423 , @xmath424 + sf : @xmath264,@xmath265,@xmath301,@xmath290,@xmath267,@xmath268,@xmath269,@xmath270,@xmath271 + gr : 1,7,19,37,61,91,127,169,217,271,331 + & ( 1450,1090)(0,130 ) ( 200,200 ) ( 1200,200)(700,1070 ) ( 45,280)@xmath237 ( 1280,280)@xmath238 ( 820,1035)@xmath239 ( 164,165)@xmath112 ( 1164,165)@xmath112 ( 664,1022)@xmath114 ( 100,100 ) ( 46,216)(100,200)(55,167 ) ( 200,300)(277,733)(613,1020 ) ( 559,1007)(613,1020)(591,969 ) ( 287,150)(700,0)(1113,150 ) ( 325,109)(287,150)(343,156 ) ( 750,983)(1150,287 ) ( 753,927)(750,983)(797,952 ) ( 650,983)(250,287 ) ( 297,318)(250,287)(253,343 ) ( 190,10)@xmath274 ( 230,700)@xmath275 ( 890,585)@xmath274 ( 680,77)@xmath275 ( 455,585)@xmath273 + @p172ptp174pt@ * automaton number @xmath425 * @p48ptp200pt @xmath426 @xmath427 @xmath245 & group : @xmath428 free of rank @xmath134 ( aleshin - vorobets ) contracting : _ no _ self - replicating : _ no _
+ rels : + sf : @xmath264,@xmath265,@xmath301,@xmath290,@xmath303,@xmath269,@xmath271,@xmath404,@xmath429 + gr : 1,7,37,187,937,4687 + & ( 1450,1090)(0,130 ) ( 200,200 ) ( 1200,200)(700,1070 ) ( 45,280)@xmath237 ( 1280,280)@xmath238 ( 820,1035)@xmath239 ( 164,165)@xmath112 ( 1164,165)@xmath112 ( 664,1022)@xmath114 ( 300,200)(1,0)800 ( 1050,225)(1100,200)(1050,175 ) ( 200,300)(277,733)(613,1020 ) ( 559,1007)(613,1020)(591,969 ) ( 1300,100 ) ( 1345,167)(1300,200)(1354,216 ) ( 750,983)(1150,287 ) ( 753,927)(750,983)(797,952 ) ( 650,983)(250,287 ) ( 297,318)(250,287)(253,343 ) ( 680,240)@xmath274 ( 230,700)@xmath275 ( 890,585)@xmath274 ( 1160,10)@xmath275 ( 455,585)@xmath273 + @p172ptp174pt@ * automaton number @xmath430 * @p48ptp200pt @xmath308 @xmath431 @xmath325 & group : @xmath432 contracting : _ yes _
self - replicating : _ yes _
+ rels : @xmath247 , @xmath248 , @xmath249 , @xmath408 , @xmath433 , + @xmath434 , @xmath435 , @xmath436 , + @xmath437 , @xmath438 , + @xmath439 , @xmath440 , + @xmath441 + sf : @xmath264,@xmath265,@xmath301,@xmath290,@xmath302,@xmath267,@xmath303,@xmath268,@xmath315 + gr : 1,4,10,19,31,46,64,85,109,136,166 + limit space : @xmath221-dimensional sphere @xmath442 & ( 1450,1090)(0,130 ) ( 200,200 ) ( 1200,200)(700,1070 ) ( 45,280)@xmath237 ( 1280,280)@xmath238 ( 820,1035)@xmath239 ( 164,165)@xmath112 ( 1164,165)@xmath112 ( 664,1022)@xmath114 ( 200,300)(277,733)(613,1020 ) ( 559,1007)(613,1020)(591,969 ) ( 287,150)(700,0)(1113,150 ) ( 325,109)(287,150)(343,156 ) ( 650,983)(250,287 ) ( 297,318)(250,287)(253,343 ) ( 1200,300)(1123,733)(787,1020 ) ( 1216,354)(1200,300)(1167,345 ) ( 150,700)@xmath273 ( 650,87)@xmath273 ( 1115,700)@xmath274(460,585)@xmath275 + @p172ptp174pt@ * automaton number @xmath443 * @p48ptp200pt @xmath320 @xmath422 @xmath373 & group : contracting : _ no _ self - replicating : _ yes _ + rels : @xmath376 , @xmath377 , @xmath378 , + @xmath379 , @xmath380 , @xmath383 , + @xmath384 , @xmath385 , + @xmath387 , @xmath388 , + @xmath389 , @xmath390 , + @xmath392 , @xmath393 , + @xmath394 + sf : @xmath264,@xmath265,@xmath266,@xmath303,@xmath327,@xmath347,@xmath348,@xmath349,@xmath350 + gr : 1,7,33,143,602,2514 + & ( 1450,1090)(0,130 ) ( 200,200 ) ( 1200,200)(700,1070 ) ( 45,280)@xmath237 ( 1280,280)@xmath238 ( 820,1035)@xmath239 ( 164,165)@xmath112 ( 1164,165)@xmath112 ( 664,1022)@xmath114 ( 100,100 ) ( 46,216)(100,200)(55,167 ) ( 300,200)(1,0)800 ( 1050,225)(1100,200)(1050,175 ) ( 287,150)(700,0)(1113,150 ) ( 325,109)(287,150)(343,156 ) ( 750,983)(1150,287 ) ( 753,927)(750,983)(797,952 ) ( 650,983)(250,287 ) ( 297,318)(250,287)(253,343 ) ( 700,1211 ) ( 820,1168)(771,1141)(779,1196 ) ( 680,240)@xmath274 ( 193,10)@xmath275 ( 890,585)@xmath274 ( 680,77)@xmath275 ( 545,1261)@xmath274 ( 460,585)@xmath275 + @p172ptp174pt@ * automaton number @xmath444 * @p48ptp200pt @xmath421 @xmath406 @xmath445 & group : isomorphic to @xmath446 contracting : _ no _ self - replicating : _ yes _
+ rels : @xmath239 , @xmath447 + sf : @xmath264,@xmath265,@xmath266,@xmath267,@xmath270,@xmath282,@xmath283,@xmath284,@xmath285,@xmath448 + gr : 1,5,17,53,161,485,1445 + & ( 1450,1090)(0,130 ) ( 200,200 ) ( 1200,200)(700,1070 ) ( 45,280)@xmath237 ( 1280,280)@xmath238 ( 820,1035)@xmath239 ( 164,165)@xmath112 ( 1164,165)@xmath112 ( 664,1022)@xmath114 ( 100,100 ) ( 46,216)(100,200)(55,167 ) ( 200,300)(277,733)(613,1020 ) ( 559,1007)(613,1020)(591,969 ) ( 287,150)(700,0)(1113,150 ) ( 325,109)(287,150)(343,156 ) ( 1300,100 ) ( 1345,167)(1300,200)(1354,216 ) ( 700,1211 ) ( 820,1168)(771,1141)(779,1196 ) ( 190,10)@xmath274 ( 230,700)@xmath275 ( 1155,10)@xmath274 ( 680,77)@xmath275 ( 465,1261)@xmath273 + @p172ptp174pt@ * automaton number @xmath449 * @p48ptp200pt @xmath308 @xmath406 @xmath445 & group : @xmath450 contracting : _ yes _ self - replicating : _ yes _ + rels :
@xmath239 , @xmath247 , @xmath451 + sf : @xmath264,@xmath265,@xmath301,@xmath266,@xmath302,@xmath268,@xmath271,@xmath347,@xmath348 + gr : 1,4,10,22,46,94,190,375,731,1422,2752,5246,9908 + limit space : & ( 1450,1090)(0,130 ) ( 200,200 ) ( 1200,200)(700,1070 ) ( 45,280)@xmath237 ( 1280,280)@xmath238 ( 820,1035)@xmath239 ( 164,165)@xmath112 ( 1164,165)@xmath112 ( 664,1022)@xmath114 ( 200,300)(277,733)(613,1020 ) ( 559,1007)(613,1020)(591,969 ) ( 287,150)(700,0)(1113,150 ) ( 325,109)(287,150)(343,156 ) ( 1300,100 ) ( 1345,167)(1300,200)(1354,216 ) ( 700,1211 ) ( 820,1168)(771,1141)(779,1196 ) ( 150,700)@xmath273 ( 1155,10)@xmath274 ( 680,77)@xmath275 ( 465,1261)@xmath273 +
we start this section with a few useful observations , which simplify computations and were used often in the classification process .
[ thm_class2states ] there are , up to isomorphism , @xmath14 different groups generated by @xmath221-state automata over a @xmath221-letter alphabet ( automata of complexity @xmath13 ) .
namely , trivial group , @xmath22 , @xmath23 , infinite cyclic group @xmath29 , infinite dihedral group @xmath452 and the lamplighter group @xmath1 .
[ nontors ] let @xmath63 be a group generated by an automaton @xmath79 over the @xmath221-letter alphabet @xmath209 that has the following property .
the set of states of @xmath79 splits into two nonempty parts @xmath205 and @xmath82 such that 1 .
one of the parts contains all active states and the other contains all inactive states ; 2 . for each state
in part @xmath205 , both arrows go to states in the same part ( either both to @xmath205 pr both to @xmath82 ) ; 3 . for each state in part @xmath82
, one arrow goes to part @xmath205 and the other to part @xmath82 then any element of the group that can be written as a product of odd number of active generators and odd number of inactive generators in any order , has infinite order . in particular
, @xmath63 is not a torsion group .
let @xmath64 be such an element .
let us prove by induction on @xmath11 that , for each @xmath160 , there exists a vertex @xmath41 fixed by @xmath453 such that the section of @xmath453 at @xmath41 has the same form ( i.e. is a product of odd number of active generators and odd number of inactive generators ) . for @xmath454
this is true .
suppose it is true for @xmath455 and let @xmath66 be a section of @xmath456 at some vertex @xmath128 satisfying the conditions of the assumption . since @xmath66 is a product of odd number of active states and odd number of inactive states we can write @xmath457 .
since @xmath41 is fixed under @xmath458 we have that @xmath459 is fixed under @xmath460 and @xmath461 .
the element @xmath462 is product ( in some order ) of the first level sections of the generators ( and/or their inverses ) used to express @xmath66 . by assumption , among these generators , there are odd number of active and odd number of inactive .
the generators from part @xmath205 , by condition ( @xmath463 ) , will produce even number of active and even number of inactive generators , while the generators from part @xmath82 , by condition @xmath464 , will give odd number of generators from both categories , which proves the induction step .
thus @xmath465 for all @xmath11 .
the abelianization of @xmath466 is isomorphic to the infinite cartesian product @xmath467 .
the canonical isomorphism sends @xmath125 to @xmath468 , where @xmath469 is the number of vertices @xmath470 , such that @xmath471 acts nontrivially on the first level ( i.e. @xmath469 is the number of active sections of @xmath64 at level @xmath228 ) [ alg_trans ] let @xmath63 be a self - similar group generated by a finite automaton over the @xmath221-letter alphabet @xmath209 , and let @xmath125 be given as a product of states and their inverses
. denote by @xmath475 the image of @xmath64 in @xmath472 $ ] .
the element @xmath64 acts level transitively on @xmath37 if and only if @xmath476 .
suppose @xmath477 , @xmath478 .
then @xmath479 we can produce similar equations for the sections @xmath480 , @xmath481 and so on .
since @xmath63 is generated by a finite automaton the number of different sections of @xmath64 is finite .
therefore we get finite linear system of equations over the variables @xmath482 , whose solution will express @xmath475 as a rational function @xmath483 , where @xmath484 are polynomials with degrees not higher than @xmath485 . expanding this rational function as a power series will produce a preperiodic sequence of coefficients from @xmath22 with period and preperiod no longer than @xmath486 .
in particular , @xmath64 acts level transitively if and only if all @xmath469 , @xmath487 are equal to @xmath114 . for the converse , let us first prove that all level stabilizers @xmath488 are different . for this
it suffices to show that for every @xmath489 @xmath490 .
since all stabilizers have finite indices in @xmath63 and @xmath63 is infinite we get that all of them are infinite .
let @xmath491 be an arbitrary nontrivial element and @xmath492 be a word of shortest length ( one of them ) such that @xmath493 ( in other words , @xmath494 and such @xmath495 is maximal ) .
clearly @xmath496 and we can consider the section @xmath497 which is an element of @xmath63 because of self - similarity .
the fact that @xmath498 implies @xmath499 .
on other hand @xmath500 .
therefore @xmath501 , thus @xmath502 and we found the desired element .
suppose @xmath63 acts transitively on level @xmath11 .
let @xmath503 be an arbitrary element and let @xmath504 be one of the words such that @xmath505 , where @xmath506 .
for @xmath507 , let us find an element @xmath125 such that @xmath508 .
this will prove the induction step .
note , that the last proof works also for self - similar subgroups of the infinitely iterated permutational wreath product @xmath516 ( the subgroup of @xmath44 consisting of those automorphisms of the @xmath38-ary tree for which the activity at every vertex is a power of some fixed cycle of length @xmath38 .
also , certain generalizations of this method could be used in more complex situations ( see , for example , the construction of @xmath517 in @xcite , or proof of transitivity of sushchansky groups on a subtree in @xcite ) .
all generators have order @xmath221 . the elements @xmath520 and @xmath521 generate @xmath30 because @xmath522 is the adding machine and has infinite order .
also we have @xmath523 and @xmath524 is normal in @xmath525 because @xmath526 and @xmath527 . in other words , @xmath528 .
furthermore , the element @xmath541 has infinite order , fixes the vertex @xmath542 and its section at this vertex is equal to @xmath541 .
therefore @xmath535 is not contracting ( all powers of @xmath541 would have to belong to the nucleus ) .
it follows from the relations @xmath544 that @xmath543 is a homomorphic image of @xmath26 . since @xmath545 , @xmath546 and @xmath547
, it follows that @xmath548 .
one can verify directly that @xmath239 is not equal to any of the four elements in @xmath549 that stabilize level 1 ( namely @xmath114 , @xmath238 , @xmath550 and @xmath551 ) .
thus @xmath552 . for a word @xmath53 in @xmath554 ,
let @xmath555 , @xmath556 and @xmath557 denote the sum of the exponents of @xmath237 , @xmath238 and @xmath239 in @xmath53 .
let @xmath53 represents the element @xmath123 .
if @xmath555 and @xmath556 are odd , then @xmath64 acts transitively on the first level , and @xmath558 is represented by a word @xmath559 , which is the product ( in some order ) of all first level sections of all generators appearing in @xmath53 .
hence , @xmath560 and @xmath561 are odd again .
therefore , similarly to proposition [ nontors ] , any such element has infinite order .
in particular @xmath562 has infinite order .
since @xmath563 and @xmath564 , the element @xmath565 has infinite order ( and so does @xmath237 ) .
since @xmath565 fixes the vertex @xmath539 and its section at that vertex is equal to @xmath565 , the group @xmath232 is not contracting . since @xmath566 is finitely generated , abelian , and self - replicating ( easy to check ) , it follows from @xcite that it is free abelian .
there are two options : either it has rank @xmath114 or rank @xmath221 ( since @xmath567 ) .
let us prove that the rank is @xmath221 .
for this it is sufficient to show that @xmath568 in @xmath63 .
assume on the contrary that @xmath569 for some integer @xmath11 and @xmath10 and choose such integers with minimal @xmath570 .
since @xmath571 stabilizes level 1 , @xmath10 must be even and we have @xmath572 .
but then @xmath573 and by the minimality assumption @xmath10 must be @xmath113 , implying @xmath574 .
the last equality can only be true for @xmath454 since @xmath566 is torsion free ( free abelian ) and @xmath575 .
thus @xmath576 .
we have @xmath578 , @xmath579 and @xmath580 .
hence , for the subgroup @xmath581 , we have @xmath582 . on the other hand
@xmath530 is normal in @xmath63 since @xmath583 and @xmath584 .
thus @xmath585 , where @xmath22 is generated by @xmath237 and the action of @xmath237 on @xmath530 is given above .
it is proved below ( see @xmath586 ) , that @xmath587 .
therefore @xmath577 also contains a torsion free subgroup of index @xmath80 .
all generators have order @xmath221 and @xmath237 commutes with @xmath239 . conjugating this group by the automorphism @xmath589 yields an isomorphic group generated by the 4-state automaton defined by the recursive relations @xmath590 , @xmath591 and @xmath592 . on the other hand , we obtain the same automaton after conjugating @xmath577 by @xmath593 ( here @xmath237 denotes the generator of @xmath577 ) .
since @xmath566 is finitely generated , abelian , and self - replicating , it follows from @xcite that it is free abelian .
consider the @xmath601-endomorphism @xmath602 associated to the vetrex @xmath113 , given by @xmath603 for @xmath604 , provided @xmath605 .
consider also the linear map @xmath606 induced by @xmath607 .
it has the following matrix representation with respect to the basis corresponding to the triple @xmath608 : @xmath609 its characteristic polynomial @xmath610 has three distinct complex roots @xmath611 , @xmath612 and @xmath613 .
choose an eigenvector @xmath614 associated to the eigenvalue @xmath615 , @xmath215 .
in particular , we may choose @xmath616 , which shows that @xmath617 in @xmath600 . in order to show that @xmath618 ( except when @xmath619 ) we will prove that the vector @xmath620 is eventually pushed out from the domain corresponding to the first level stabilizer , i.e. from the set @xmath621 , by iterations of the action of @xmath79 . consider the expansion of @xmath41 in the basis @xmath622 : @xmath623 .
since @xmath624 or @xmath625 , @xmath41 is not a scalar multiple of @xmath626 .
we have @xmath627 , as @xmath628 , since @xmath629 .
we can choose a neighborhood of @xmath630 that does not contain points from @xmath631 , except maybe @xmath630 .
eventually @xmath632 will be in this neighborhood and , since @xmath633 for all @xmath634 , @xmath632 will be outside of @xmath631 .
this implies that the word @xmath635 represents a nontrivial element in @xmath600 .
thus @xmath636 . *
846 * : @xmath326 .
this is a result of muntyan .
see the proof in @xcite .
in particular , @xmath637 contains a self - similar free group of rank @xmath221 generated by a 6-state automaton .
the automaton 846 is sometimes called bellaterra automaton . *
852 * : basilica group @xmath638 .
first studied in @xcite , where it is proved that @xmath6 is not in the class @xmath639 of sub - exponentially amenable groups , does not contain a free subgroup of rank @xmath221 , and the nontrivial generators @xmath237 and @xmath238 generate a free subsemigroup .
spectral properties are considered in @xcite .
it is proved in @xcite that @xmath6 is amenable , providing the first example of an amenable group not in the class @xmath639 .
let us prove that @xmath238 has infinite order .
for any @xmath641 @xmath642 . since @xmath643 equals either @xmath238 or @xmath239 and @xmath644 , we have @xmath645 . therefore all elements in the forward orbit of @xmath646 under the action of @xmath238 end in @xmath647 .
the length of the non - zero prefix of any infinite word ending in @xmath647 can not decrease under the action of @xmath238 .
indeed , for any @xmath641 @xmath648 . the section @xmath649 is one of the three generators , for which we have @xmath650 , @xmath651 and @xmath652 .
on the other hand , the length of the non - zero prefix along the orbit can not stabilize , because in this case the orbit must be finite and we must have @xmath653 , for some @xmath654 .
but this is impossible since @xmath655 and thus the length of the non - zero prefix of @xmath656 must be at least @xmath134 .
thus the orbit is infinite and @xmath238 has infinite order . using the same approach as for @xmath640 one can show that @xmath239 has infinite order .
namely the length of the non - zero prefix of the forward orbit of @xmath658 under @xmath239 is nondecreasing , which then implies that this orbit is not finite .
the automaton satisfies the conditions of proposition [ nontors ] .
thus , @xmath660 has infinite order in @xmath63 , which implies that @xmath661 , @xmath662 also have infinite order . hence , we can claim the same for @xmath237 and @xmath360 .
furthermore , the element @xmath663 also has infinite order ( it is conjugate of the adding machine ) .
since @xmath664 we have @xmath665 and @xmath666 .
let us check that @xmath667 .
since @xmath668 and @xmath669 all we need to check is that @xmath670 , i.e. @xmath671 . the last is correct since @xmath672 and @xmath673 is a conjugate of the inverse of @xmath674 .
thus , since @xmath238 and @xmath675 have infinite order , @xmath676 .
denote @xmath678 and @xmath679 .
all generators have order @xmath221 , and therefore the subgroup @xmath680 is a normal subgroup of index @xmath221 in @xmath677 .
moreover @xmath681 , where @xmath22 is generated by @xmath239 and the action of @xmath22 on @xmath530 is given by @xmath682 and @xmath683 .
we have @xmath684 and @xmath685 . exchanging the letters @xmath113 and @xmath114 leads to an isomorphic copy of @xmath530 defined by @xmath686 and @xmath687 , which is the iterated monodromy group @xmath688 , according to @xcite .
thus , @xmath689 .
it is known @xcite that the group @xmath692 defined by @xmath693 is the lamplighter group @xmath1 ( compare to ) .
consider the subtree @xmath694 of @xmath37 consisting of all words over the alphabet @xmath695 .
the element @xmath237 swaps the letters of @xmath151 and @xmath238 fixes them .
since @xmath696 , @xmath697 , the tree @xmath694 is invariant under the action of @xmath698 and the action of @xmath530 on @xmath694 coincides with the action of the lamplighter group @xmath699 on @xmath37 ( with the identification @xmath700 ) .
therefore the map @xmath701 given by @xmath702 , @xmath703 extends to a homomorphism .
we claim that this homomorphism has trivial kernel .
indeed , let @xmath704 be a group word representing an element of the kernel of @xmath607 .
since the word @xmath705 represents the identity in @xmath706 the total exponent of @xmath237 in @xmath53 must be even and the total exponent @xmath707 of both @xmath237 and @xmath238 in @xmath53 must be 0 .
but in that case the element @xmath708 fixes the top two levels of the tree @xmath37 and has decomposition @xmath709 where the @xmath710 s denote words over @xmath237 and @xmath238 representing the identity in @xmath530 ( these words correspond to the first level sections of @xmath705 in @xmath706 ) .
therefore @xmath711 and the kernel of @xmath607 is trivial .
this group contains @xmath32 as a subgroup of index @xmath397 .
it is contracting with nucleus consisting of @xmath713 elements , whose self - similar closure consists of @xmath714 elements .
all generators have order @xmath221 .
let @xmath715 , @xmath716 , and consider the subgroup @xmath717 .
direct computations show that @xmath50 and @xmath65 commute ( @xmath718 and @xmath719 ) .
conjugating by @xmath720 leads to the self - similar copy @xmath721 of @xmath142 generated by @xmath722 and @xmath723 , where @xmath724 and @xmath725 . since @xmath726
, the virtual endomorphism of @xmath721 is given by the matrix @xmath727 the eigenvalues @xmath728 of this matrix are not algebraic integers , hence , according to @xcite , the group @xmath721 is free abelian of rank @xmath221 , and so is @xmath142 . since all generators have order @xmath221 , the subgroup @xmath729 has index @xmath221 in @xmath712 .
the stabilizer @xmath730 of the second level has index @xmath731 in @xmath530 .
moreover , the quotient group is isomorphic to the dihedral group @xmath24 ( since @xmath660 acts on the second level by permuting @xmath732 and @xmath733 , while @xmath734 acts by permuting @xmath735 ) .
the stabilizer @xmath730 , conjugated by the element @xmath736 , is generated by @xmath737 therefore , all @xmath738 commute and @xmath739 .
if @xmath740 , then all sections must be trivial , hence , @xmath741 .
but @xmath142 is free abelian , whence @xmath742 , @xmath743 .
thus , @xmath730 is a free abelian group of rank @xmath744 . since @xmath748 and @xmath749 , the generators @xmath237 and @xmath239 commute
further , @xmath750 , which shows that @xmath751 , and therefore @xmath752 and @xmath753 .
since @xmath754 , the element @xmath237 has infinite order and so does @xmath755 .
finally , since @xmath756 , we have @xmath757 and @xmath747 is the klein bottle group .
going back to the generating set consisting of @xmath237 and @xmath238 , we get the presentation @xmath758 . *
2240 * : free group of rank @xmath134 .
the automaton generating this group first appeared in @xcite .
it is proved in @xcite that @xmath759 is a free group of rank 3 with basis @xmath608 .
this is the smallest example of a free nonabelian group among all automata over a @xmath221-letter alphabet ( see theorem [ thm : class_free ] ) .
all generators have order @xmath221 .
let @xmath762 and @xmath763 and let @xmath764 . then @xmath765 and @xmath766 .
it is easy to check that @xmath50 and @xmath65 commute and that @xmath530 is self - replicating .
the matrix of the associated virtual endomorphism is given by @xmath767 since the eigenvalues @xmath768 are not algebraic integers , according to @xcite @xmath530 is free abelian of rank 2 .
the subgroup @xmath530 is normal of index 2 in @xmath761 because the generators of @xmath761 are of order @xmath221 .
thus @xmath769 , where the action of @xmath770 on @xmath530 is by inversion of the generators .
for any vertex @xmath128 , we have @xmath772 , @xmath773 and @xmath774 , for @xmath775 .
therefore , for any vertex @xmath776 , @xmath777 and the forward orbit of @xmath658 under @xmath237 is infinite , because the length of the non - zero prefix grows by 2 with each application of @xmath237 .
thus @xmath237 has infinite order .
the subsemigroup @xmath787 is free .
indeed , let @xmath53 be a nonempty word in @xmath788 .
if @xmath789 in @xmath690 , then @xmath53 contains both @xmath237 and @xmath238 , because they both have infinite order .
suppose the length of @xmath53 is minimal among all nonempty words over @xmath790 representing the identity element in @xmath690 .
then one of the projections of @xmath53 will be shorter than @xmath53 , nonempty , and will represent the identity in @xmath690 , which contradicts the minimality assumption .
thus @xmath791 in @xmath690 , for any nonempty word in @xmath788 . 1 .
if @xmath53 ends in @xmath794 then @xmath559 is a word that is shorter than @xmath53 ending in @xmath237 , while @xmath795 is a word not longer than @xmath41 ending in @xmath238 . since @xmath796 in @xmath690 and @xmath797
, we have a contradiction .
if @xmath53 ends in @xmath798 then @xmath799 is a word shorter than @xmath53 ending in @xmath238 , while @xmath626 is a word not longer than @xmath41 ending in @xmath237 . since @xmath800 in @xmath690 and @xmath801
, we have a contradiction .
3 . if @xmath802 then @xmath803 in @xmath63 and @xmath626 is a nonempty word , which is impossible , as already proved above .
it is proved in @xcite that @xmath450 is generated by @xmath808 and @xmath809 .
we have then @xmath810 .
conjugate the right hand side of the wreath recursion by @xmath811 to obtain a copy of @xmath450 given by @xmath812 , @xmath813 and @xmath814 ( this is equivalent to conjugating by @xmath815 in @xmath816 ) .
this shows that @xmath807 is isomorphic to @xmath450 via the isomorphism @xmath817 and @xmath818 .
moreover , they are conjugate by the element @xmath819 , where @xmath820 ( this is the automorphism of the tree changing all letters which stand on even places ) . the group @xmath807 is contained in @xmath577 as a subgroup of index @xmath221 ( see @xmath577 ) .
it contains the torsion free subgroup @xmath530 mentioned in the discussion of @xmath577 as subgroup of index 2 and is a weakly branch group over @xmath596 .
all schreier graphs on the boundary of the tree have polynomial growth of degree @xmath221
. diameters of schreier graphs on the levels grow as @xmath822 ( see @xcite for details ) .
laurent bartholdi and rostislav i. grigorchuk .
lie methods in growth of groups and groups of finite width . in michael atkinson et al . , editor , _ computational and geometric aspects of modern algebra _ , volume 275 of _ london math .
_ , pages 127 .
cambridge univ . press ,
cambridge , 2000 .
laurent i. bartholdi and zoran unik . some solvable automaton groups . in _
topological and asymptotic aspects of group theory _ ,
volume 394 of _ contemp .
_ , pages 1129 .
soc . , providence , ri , 2006 .
steven c. ferry , andrew ranicki , and jonathan rosenberg . a history and survey of the novikov conjecture . in _
novikov conjectures , index theorems and rigidity , vol . 1 ( oberwolfach , 1993 ) _ , volume 226 of _ london math .
lecture note ser .
_ , pages 766 .
cambridge univ . press ,
cambridge , 1995 .
rostislav i. grigorchuk and andrzej uk .
spectral properties of a torsion - free weakly branch group defined by a three state automaton . in _
computational and statistical group theory ( las vegas , nv / hoboken , nj , 2001 ) _ , volume 298 of _ contemp .
_ , pages 5782 .
soc . , providence , ri , 2002 .
wolfgang lck . , volume 44 of _ ergebnisse der mathematik und ihrer grenzgebiete .
3 . folge . a series of modern surveys in mathematics [ results in mathematics and related areas .
3rd series .
a series of modern surveys in mathematics]_. springer - verlag , berlin , 2002 .
v. nekrashevych and s. sidki .
automorphisms of the binary tree : state - closed subgroups and dynamics of @xmath825-endomorphisms .
volume 311 of _ london math .
_ , pages 375404 .
cambridge univ . press , 2004 . | an approach to a classification of groups generated by 3-state automata over a 2-letter alphabet and the current progress in this direction are presented .
several results related to the whole class are formulated . in particular , all finite , abelian , and free groups are classified .
in addition , we provide detailed information and complete proofs for several groups from the class , with the intention of showing the main methods and techniques used in the classification . | [
2,
145,
933,
1412,
1746,
113,
1211,
113,
21109,
60110,
304,
110,
107,
106,
253,
1746,
127,
110,
108,
211,
113,
149,
110,
108,
813,
233,
23956,
110,
108,
15618,
1746,
110,
108,
111,
114,
14846,
13783,
130,
1211,
4216,
124,
11723,
2115,
110,
107,
106,
237,
110,
108,
145,
403,
120,
1211,
113,
60110,
304,
2885,
109,
1444,
113,
109,
3277,
113,
813,
233,
984,
1211,
111,
2332,
110,
108,
111,
127,
985,
112,
109,
692,
113,
129,
445,
457,
63100,
116,
111,
7248,
116,
6870,
8909,
2073,
12788,
113,
31289,
144,
79597,
18618,
2650,
757,
46819,
110,
107,
106,
145,
163,
403,
120,
1211,
113,
60110,
304,
137,
129,
263,
112,
692,
743,
113,
110,
73058,
208,
4063,
1451,
304,
39293,
5492,
110,
107,
106,
1477,
110,
108,
145,
319,
181,
909,
602,
985,
112,
109,
755,
113,
55127,
21109,
60110,
304,
110,
107,
1,
0,
0,
0,
0,
0,
0,
0,
0,
0,
0,
0,
0,
0,
0,
0,
0,
0,
0,
0,
0,
0,
0,
0,
0,
0,
0,
0,
0,
0,
0,
0,
0,
0,
0,
0,
0,
0,
0,
0,
0,
0,
0,
0,
0,
0,
0,
0,
0,
0,
0,
0,
0,
0,
0,
0,
0,
0,
0,
0,
0,
0,
0,
0,
0,
0,
0,
0,
0,
0,
0,
0,
0,
0,
0,
0,
0,
0,
0,
0,
0,
0,
0,
0,
0,
0,
0,
0,
0,
0,
0,
0,
0,
0,
0,
0,
0,
0,
0,
0,
0,
0,
0,
0,
0,
0,
0,
0,
0,
0
] |
"in the early universe , a scalar field often oscillates coherently after inflation @xcite .\ncoher(...TRUNCATED) | " cosmological dynamics of scalar field with a monomial potential @xmath0 with a general background (...TRUNCATED) | [2,114,61388,764,432,76281,116,20141,445,244,7615,115,109,616,5447,110,107,115,136,800,110,108,145,6(...TRUNCATED) |
"kouveliotou ( 1993 ) showed that the durations of gamma ray bursts ( grbs ) have a bimodal distribu(...TRUNCATED) | " if the efficiency for producing @xmath0-rays is the same in short duration ( @xmath1 s ) gamma - r(...TRUNCATED) | [2,244,42225,15291,135,613,111,300,35421,20610,27305,143,17050,12903,110,158,148,167,571,174,9529,11(...TRUNCATED) |
"according to our current understanding @xcite , dark matter ( dm ) and dark energy ( de ) represent(...TRUNCATED) | " scalar - tensor theories of gravity provide a consistent framework to accommodate an ultra - light(...TRUNCATED) | [2,126,148,174,1673,120,61388,233,77076,9941,143,11607,110,158,3277,2650,757,46819,137,763,114,1443,(...TRUNCATED) |
"the conception and construction of systems of well - defined coupled macrospins underpins both the (...TRUNCATED) | " we report simulations of a frustrated odd - numbered macrospin ring system , with full point dipol(...TRUNCATED) | [2,2573,1645,115,109,633,113,12587,40401,6936,110,108,9736,61420,111,5588,37801,133,149,174,4125,303(...TRUNCATED) |
"the carina dwarf spheroidal ( dsph ) galaxy has an important place in the history of our understand(...TRUNCATED) | " we have discovered 20 dwarf cepheids ( dc ) in the carina dsph galaxy from the analysis of individ(...TRUNCATED) | [2,109,439,6284,23832,110,116,30713,71508,143,3138,116,9342,110,158,15187,117,142,459,110,108,10545,(...TRUNCATED) |
"the interest in strongly correlated systems in frustrated lattices has increased recently because o(...TRUNCATED) | " we discuss the magnetic phases of the hubbard model for the honeycomb lattice both in two and thre(...TRUNCATED) | [2,109,6451,49556,861,115,109,42981,31971,117,10920,118,109,8357,40401,3181,113,109,575,2650,757,468(...TRUNCATED) |
"homework is a key element in every undergraduate - level physics course for practicing problem solv(...TRUNCATED) | " the mathematization project investigates students use of mathematical tools across the undergradua(...TRUNCATED) | [2,126,117,356,112,630,199,392,1138,10045,112,1113,8343,6727,743,110,107,115,136,800,110,108,145,207(...TRUNCATED) |
End of preview. Expand
in Dataset Viewer.
Dataset Card for AutoTrain Evaluator
This repository contains model predictions generated by AutoTrain for the following task and dataset:
- Task: Summarization
- Model: google/bigbird-pegasus-large-pubmed
- Dataset: scientific_papers
To run new evaluation jobs, visit Hugging Face's automatic model evaluator.
Contributions
Thanks to @lewtun for evaluating this model.
- Downloads last month
- 36