singular continuous spectrum for palindromic schrodinger operators we give new examples of discrete schrodinger operators with potentials taking finitely many values that have purely singular continuous spectrum if the hullx of the potential is strictly ergodic then the existence of just one potentialx inx for which the operator has no eigenvalues implies that there is a generic set inx for which the operator has purely singular continuous spectrum a sufficient condition for the existence of such anx is that there is az x that contains arbitrarily long palindromes thus we can define a large class of primitive substitutions for which the operators are purely singularly continuous for a generic subset inx the class includes well known substitutions like fibonacci thue morse period doubling binary non pisot and ternary non pisot we also show that the operator has no absolutely continuous spectrum for allx x ifx derives from a primitive substitution for potentials defined by circle maps x n 1 j 0 n we show that the operator has purely singular continuous spectrum for a generic subset inx for all irrational and every half open intervalj

on the characteristic polynomials of fibonacci chains special diatomic linear chains with elastic nearest neighbour interaction and the two masses distributed according to the binary fibonacci sequence are studied

the greatest fixed points and rational omega tree languages abstract rational omega tree languages are defined by extending the notion of rational or regular expressions for languages and their properties are investigated they are characterized by means of greatest fixed points of certain functions and are shown to be precisely the omega tree languages recognized by rabin s special automata a simple normal form for rational omega tree languages is also presented

localized wavefunctions in quantum systems with multiwell potentials one dimensional systems with the multiwell potential v x 2 1cos x 2 2cos x are considered localized wavefunctions in the case when the energy is greater than the maximum value of the potential are presented wigner and weyl functions corresponding to the wavefunctions of these systems are also studied it is shown that they are very sensitive to the value of the parameter

critical properties of electron eigenstates in incommensurate systems this paper describes some properties of the eigenvalue equation n 1 n 1 2 cos 2 n n e n this is an example of the more general problem of a hermitian eigenvalue equation in the form of a difference equation with periodic coefficients these equations arise in solid state physics they occur in connection with tight binding models for electrons in one dimensional solids with an incommensurate modulation of the structure and in models for the energy bands of bloch electrons moving in a plane with a perpendicular magnetic field the model studied has a critical point when 1 following some earlier work by azbel azbel m ya phys rev lett 43 1954 1979 an approximate renormalization group transformation is derived this predicts that the spectrum and eigenstates have a remarkable recursive structure at the critical point which is dependent on the expansion of as a continued fraction also when is an irrational number there is a localization transition from extended states to localized states as increases through the critical point this localization transition which was previously discovered by aubry andre aubry s andre g ann israel phys soc 3 133 1979 using the thouless formula for the localization length is explained by the renormalization group transformation derived here

transmission fingerprints of quasi periodic chains abstract we propose a method to characterize quasi periodic chains based on the transmission probabilities we consider finite layers which are building up following the fibonaci thue morse and cantor sequences the spectra show an undoubtedly fractal behavior with a distinct appearence for each chain furthermore we construct return maps which present the same characteristic attractor for all peaks of energies within the same kind of chain

renormalisation procedure for the quasiperiodic schrodinger equation a renormalisation procedure is constructed for the one dimensional schrodinger equation with a quasiperiodic potential the renormalisation transformation has a trivial fixed point describing the behaviour of the system near the unperturbed free motion and a non trivial fixed point corresponding to the critical case a number system of an irrational base is introduced for the scaling of the spectra

phase transitions on strange sets the ising quasicrystal a quantum ising spin chain with nearest neighbour couplings arranged in a quasiperiodic sequence is considered the cantor set structure of the energy spectrum is analysed in terms of the thermodynamic description of multifractals evidence is given that the spectrum of scales develops a singular behaviour this is associated with a first order phase transition of a new type it is argued that this effect involves not only quantum spins but the whole class of phonon like propagation problems on quasiperiodic chains

influence of the anisotropy parameter δ on the spectrum of the generalized q symmetrized harper equation the 1 generalization of the q symmetrized harper equation is discussed in terms of wavefunctions expressed by laurent series proceeding by recursion leads to a nontrivial dependent generalization of the characteristic energy polynomial with a special emphasis on a continuous dependence on the commensurability parameter the multiplicity parameter which is responsible for the amount of coprime realizations of the commensurability parameter is also accounted for the present energies have been derived so as to reproduce particular ones obtained before as limiting cases

results on the propositional μ calculus abstract in this paper we define and study a propositional calculus l which consists essentially of propositional modal logic with a least fixpoint operator l is syntactically simpler yet strictly more expressive than propositional dynamic logic pdl for a restricted version we give an exponential time decision procedure small model property and complete deductive system theory subsuming the corresponding results for pdl

quantum diffusion in the generalized harper equation we study numerically the dynamic and spectral properties of a one dimensional quasi periodic system where site energies are given by with denoting the kth quasiperiodic lattice site when is given by the reciprocal lattice vector with n and m being successive fibonacci numbers the variance of the wavepacket is found to grow quadratically in time regardless of the potential strength v for other values of f there exists a critical value of v beyond which the growth of the wavepacket variance is bounded in particular an anomalous diffusion takes place for corresponding to with generic integers m and n the level spacing distribution is also examined and the corresponding exponent is observed to decrease with v

classical representation of the one dimensional anderson model a new approach is applied to the one dimensional anderson model by using a two dimensional hamiltonian map for a weak disorder this approach allows for a simple derivation of correct expressions for the localization length both at the centre and at the edge of the energy band where standard perturbation theory fails approximate analytical expressions for strong disorder are also obtained

some remarks on discrete aperiodic schrodinger operators we consider schrodinger operators onl2 with deterministic aperiodic potential and schrodinger operators on the l2 space of the set of vertices of penrose tilings and other aperiodic self similar tilings the operators onl2 fit into the formalism of ergodic random schrodinger operators hence their lyapunov exponent integrated density of states and spectrum are almost surely constant we show that they are actually constant the lyapunov exponent for one dimensional schrodinger operators with potential defined by a primitive substitution the integrated density of states and the spectrum in arbitrary dimension if the system is strictly ergodic we give examples of strictly ergodic schrodinger operators that include several kinds of almost periodic operators that have been studied in the literature for schrodinger operators on penrose tilings we prove that the integrated density of states exists and is independent of boundary conditions and the particular penrose tiling under consideration

the spectrum of a quasiperiodic schrodinger operator the spectrum h of the tight binding fibonacci hamiltonian hmn m n 1 m 1 n m n v n v n chi omega 3 omega 2 n 1 1 is the golden number is shown to coincide with the dynamical spectrum the set on which an infinite subsequence of traces of transfer matrices is bounded the point spectrum is absent for any and h is a cantor set for 4 combining this with casdagli s earlier result one finds that the spectrum is singular continuous for 16

electronic properties of the tight binding fibonacci hamiltonian the analysis of the electronic properties of the tight binding fibonacci hamiltonian is carried out using dynamical systems techniques two classes of fibonacci sequences are considered corresponding to the cases when there are two types of building blocks and also when there are three types of building blocks the recursion relations for the traces of the transfer matrices are determined and studied for various extensions of the fibonacci case with the golden mean some differences are obtained between the various types of second order fibonacci sequences and are most likely due to long range order applications of this work to the transmission of light through a multilayered medium and the electrical resistance of a one dimensional quasicrystal as determined by the landauer formula are presented

the landauer conductance of generalised fibonacci superlattices numerical results abstract the quantum model for quasi one dimensional generalised fibonacci superlattices gfs is formulated the landauer electrical conductance l of gfs is studied numerically it is shown that l as a function of incident particle energy e oscillates strongly and exhibits resonant character we have verified numerically that l e reaches its local maximum for e corresponding to energy eigenenvalues of charge in superlattices

algebraic structures for 1d quasiperiodic systems the structure of certain quasiperiodic systems is analysed and interpreted by algebraic means all systems considered have a free monoid and semigroup structure with n generators as treated in the field of combinatorics on words an extension from free monoids to free groups and their automorphisms as treated by combinatorial group theory applies to a class of iterative systems the group structures yield algebraic relations and conserved quantities examples encompass the fibonacci system and its generalizations motivated by the applications to the physics of these systems homomorphisms from the free groups into linear operators of su 2 and su 1 1 are studied the systems are also viewed in a geometry due to fricke and klein the homomorphism to the group su 1 1 is the basis for the determination of 1d electronic spectra the class types their multiplication and the commutator properties are given the physics of these systems corresponds to the continuous schrodinger equation the quasiperiodic structure gives rise to exact relations between continuous and discrete systems

symbolic dynamics for the renormalization map of a quasiperiodic schrodinger equation a rigorous analysis is given of the dynamics of the renormalization map associated to a discrete schrodinger operatorh onl 2 defined byh n n 1 n 1 vf n n wherev is a real parameter f is a certain discontinuous period 1 function and sigma left 1 sqrt 5 right mathord left vphantom left 1 sqrt 5 right 2 right kern nulldelimiterspace 2 is the golden mean the renormalization map forh is a diffeomorphism t of 3 preserving a cubic surfaces v forv 8 we prove that the non wandering set of the restriction oft tos v is a hyperbolic set on whicht is conjugate to a subshift on six symbols it follows from results in dynamical systems theory that the optimally approximating periodic operators toh have spectra which obey a global scaling law we also define a set which we call the pseudospectrum of the operatorh we prove it to be a cantor set of measure zero and obtain bounds on its hausdorff dimension it is an open question whether the pseudospectrum coincides with the spectrum ofh

vibrations of simple fractal based models dynamical systems with fractal geometry can be constructed in a variety of ways we illustrate this variety with examples based on the cantor set the sierpinski gasket and on lattices of these fractal based structures depending on the physical parameters the models can exhibit both discrete and continuous spectra

intermittency in fibonacci chains the devil s staircase spectrum of a fibonacci chain is obtained using a simple non linear map this map exhibits intermittency for all fibonacci number iterates of the function whenever the frequency lies in a gap of the spectrum the self similar nature of the eigenfunctions is shown to be a consequence of the intermittency behaviour certain simple patterns that emerge in the gap labelling theorem are discussed

repeller structure in a hierarchical model ii metric properties we study the renormalization dynamics deriving from a hierarchical tight binding schrodinger equation in the part i of this work we analyzed the topological structure of the recurrent set a chaotic repeller and its relation with the spectral problem in this part we turn our attention to the metric properties of the repeller we first study periodic orbits and their bifurcation unfolding and we organize them on a binary tree we then apply a thermodynamic formalism which provides a complete characterization of the scaling properties of the energy spectrum the distributionf of local dimensions is determined by computing both a generalized function through the periodic orbits and the bandwidths of periodic approximants of the schrodinger operator when the growth rater of the potential is smaller than 1 we find evidence of a phase transition implying that two different classes of states coexist in the spectrum the asymptotic behavior of the lebesgue measure of the spectrum is also studied a linear scaling of to 0 is observed forr 1 while forr 1 the measure of the periodic approximants goes to 0 as r h with the hierarchical orderh finally we show that the localized state present for r 1 is characterized by a superexponential scaling of the bandwidth

localization problem in one dimension mapping and escape a one dimensional schr odinger equation in a discontinuous quasiperiodic potential is reduced to a recursion relation for transfer matrices and then to one for traces of these matrices when the potential is periodic the bandwidth goes to zero as an algebraic function of the period with a critical index which depends upon the potential strength this critical index is also evaluated as the solution to an escape rate problem for the recursion relations

statistics of resonances and delay times in random media beyond random matrix theory we review recent developments in quantum scattering from mesoscopic systems various spatial geometries whose closed analogues show diffusive localized or critical behaviour are considered these are the features that cannot be described by the universal random matrix theory results instead one has to go beyond this approximation and incorporate them in a non perturbative way here we pay particular attention to the traces of these non universal characteristics in the distribution of the wigner delay times and resonance widths the former quantity captures time dependent aspects of quantum scattering while the latter is associated with the poles of the scattering matrix

critical points for finite fibonacci chains of point delta interactions and orthogonal polynomials for a one dimensional schrodinger operator with a finite number n of point delta interactions with a common intensity the parameters are the intensity the n 1 intercenter distances and the mass critical points are points in the parameters space of the hamiltonian where one bound state appears or disappears the study of critical points for hamiltonians with point delta interactions arranged along a fibonacci chain is shown to be closely related to the study of the so called fibonacci operator a discrete one dimensional schrodinger type operator which occurs in the context of tight binding hamiltonians these critical points are the zeros of orthogonal polynomials previously studied in the context of special diatomic linear chains with elastic nearest neighbor interaction properties of the zeros location asymptotic behavior gaps are investigated the perturbation series from the solvable periodic case is determined the measure which yields orthogonality is investigated numerically from the zeros it is shown that the transmission coefficient at zero energy can be expressed in terms of the orthogonal polynomials and their associated polynomials in particular it is shown that when the number of point delta interactions is equal to a fibonacci number minus 1 i e when the intervals between point delta interactions form a palindrome all the fibonacci chains at critical points are completely transparent at zero energy

the periodically kicked rotator recurrence and or energy growth we explore the properties of the quantum kicked rotator its classical equivalent being the standard map its behavior as found by computer studies depends very much on the strength of the external forcing at low strength it is seemingly recurrent in the sense of hogg and huberman however its energy increases with time at large forcings for quantum systems a unitary map defines the evolution over one period of time the spectrum of this map in an infinite space does not seem to change continuously when one approaches the ratio of the frequencies of the external and of the unperturbed system by rational approximations of the golden mean

remarks on the spectral properties of tight binding and kronig penney models with substitution sequences we comment on some recent investigations on the electronic properties of models associated with the thue morse chain point out that their conclusions contradict rigorously proven theorems and indicate some of the sources of these misinterpretations we briefly review and explain the current status of mathematical results in this field and discuss some conjectures and open problems

dynamical system related to quasiperiodic schrodinger equations in one dimension a dynamical map is obtained from a class of quasiperiodic discrete schrodinger equations in one dimension which include the fibonacci system the potentials are constant except for steps at special points

renormalization group approach to the local green functions of a family of generalized fibonacci lattices we study the local electronic properties of a family of generalized fibonacci lattices associated with the sequences which are given by the inflation rule a b ab n a where n is an arbitrary positive integer greater than one a unified real space renormalization group approach is presented to calculated the local green function and the local density of states at any given site

one dimensional markovian field ising model physical properties and characteristics of the discrete stochastic mapping the zero temperature physical properties of an ising chain in a markovian field taking only two values with non zero mean are evaluated and a related discrete stochastic mapping with the theory of finite markov chains is investigated a discontinuous behaviour of the magnetisation and the residual entropy dependent on both mean field and exchange is found which can be related to flips of microscopic spin clusters for non zero temperature the mapping is characterised by the fractal properties of its attractor and by the lyapunov exponent an explicit expression for the measure in the nth iteration of the chapman kolmogorov equation is obtained

dynamical upper bounds on wavepacket spreading we derive a general upper bound on the spreading rate of wavepackets in the framework of schrodinger time evolution our result consists of showing that a portion of the wavepacket cannot escape outside a ball whose size grows dynamically in time where the rate of this growth is determined by properties of the spectral measure and by spatial properties of solutions of an associated time independent schrodinger equation we also derive a new lower bound on the spreading rate which is strongly connected with our upper bound we apply these new bounds to the fibonacci hamiltonian the most studied one dimensional model of quasicrystals as a result we obtain for this model upper and lower dynamical bounds establishing wavepacket spreading rates which are intermediate between ballistic transport and localization the bounds have the same qualitative behavior in the limit of large coupling

on the hausdorff dimension of the spectrum of the thue morse hamiltonian we show that the hausdorff dimension of the spectrum of the thue morse hamiltonian has a common positive lower bound for all coupling

on the trace map for products of matrices associated with substitutive sequences in a recent article m kola and m k ali study the polynomial trace map for products of matrices associated with substitutive sequences on a two letter alphabet the existence of which has been proved by j p allouche and j peyriere computer calculations led them to conjecture some divisibility properties of the involved polynomials the present work explains mathematically why it is so

novel computation of the growth rate of generalized random fibonacci sequences a new formulation is proposed for the computation of average growth rates of generalized random fibonacci sequences based on the new formula a novel numerical scheme is designed and successfully implemented and interesting analytic asymptotic expansions are obtained for several examples

on kicked systems modulated along the thue morse sequence we give necessary conditions for the fourier transform of the time evolution operator of kicked systems modulated along the thue morse sequence to present dirac s delta functions the case of two level systems is analysed in detail and it is found based on the trace map and numerical investigations that generically its time evolution has no quasi periodic component

partition function zeros for aperiodic systems the study of zeros of partition functions initiated by yang and lee provides an important qualitative and quantitative tool in the study of critical phenomena this has frequently been used for periodic as well as hierarchical lattices here we consider magnetic field and temperature zeros of ising model partition functions on several aperiodic structures in 1d we analyze aperiodic chains obtained from substitution rules the most prominent example being the fibonacci chain in 2d we focus on the tenfold symmetric triangular tiling which allows efficient numerical treatment by means of corner transfer matrices

logical definability of fixed points note abstract we define a class of monotonic functions whose least fixed points are weakly definable in the sense of rabin we use this property to give a new proof that v definable sets of trees are recognized by rabin s special automata

franco american meeting on the mathematics of random and almost periodic potentials we present the abstracts of papers given at a conference held at the ecole polytechnique in june 1983 on the mathematics of random and almost periodic potentials

cantor spectrum and singular continuity for a hierarchical hamiltonian we study the spectrum of the hamiltonianh onl2 given by h n n 1 n 1 v n n with the hierarchical ultrametric potentialv 2m 2l 1 1 rm 1 r corresponding to 1 2 and 3 dimensional coulomb potentials for 0 1 respectively in a suitably chosen valuation metric we prove that the spectrum is a cantor set and gaps open at the eigenvaluesen 1 en 2 en 2n 1 of the dirichlet problemh e 0 2n 0 n 1 in the gap opening aten k the integrated density of states takes on the valuek 2n the spectrum is purely singular continuous forr 1 when the potential is unbounded and the lyapunov exponent vanishes in the spectrum the spectrum is purely continuous forr 1 in h 2 2 and 0 here but one cannot exclude the presence of eigenvalues near the border of the spectrum we also propose an explicit formula for the green s function

upper bounds in quantum dynamics of course the case of main interest is where h is given by l2 rd or 2 zd h is a schr dinger operator of the form a v and ip 0 is a localized wavepacket under the time evolution 1 the wavepacket i t will in general spread out with time finding quantitative bounds for this spreading in concrete cases and developing methods for proving such bounds has been the objective of a great number of papers in recent years one is often interested in the growth behavior of the moments of the position operator that is the function 11 x p 2 i 2 these questions are particularly complicated and interesting in situations where singular continuous spectra occur the discovery of the occurrence of such spectra for many models of physical relevance has triggered some activity on the quantum dynamical side for models with absolutely continuous spectra on the other hand quantum dynamical questions have been investigated much earlier in the context of scattering theory and have been answered to a great degree of satisfaction models with point spectrum or more restrictively models exhibiting anderson localization have also been investigated from a quantum dynamical point of view while it is not true in general that spectral localization implies dynamical localization 26 73 it has been shown that the three main approaches to spectral localization the fractional moment method 3 4 5 multi scale analysis 28 27 75 and the bourgain goldstein schlag method 12 37 yield dynamical localization as a by product 2 13 22 31 34 quantum dynamics for schr dinger operators with singular continuous spectra is much less understood the two primary examples motivated by physics are the harper model also known as the almost mathieu operator at critical cou pling which describes a bloch electron in a constant magnetic field with irrational flux through a unit cell and the fibonacci model which describes a standard one dimensional quasicrystal both models have been studied heavily in both the physics and mathematics communities

self similarity of binary quasiperiodic sequences self similarity in binary quasiperiodic sequences generated by a projection method is shown to exist when and only when it is associated with quadratic irrational numbers and the explicit self similarity transformation for an arbitrary quadratic number is obtained the self similarity transformation is shown not to be reducible to the simplest form for a class of quadratic numbers

block variables for deterministic aperiodic sequences we use the concept of block variables to obtain a measure of order disorder for some one dimensional deterministic aperiodic sequences for the thue morse sequence the rudin shapiro sequence and the period doubling sequence it is possible to obtain analytical expressions in the limit of infinite sequences for the fibonacci sequence we present some analytical results which can be supported by numerical arguments it turns out that the block variables show a wide range of different behaviour some of them indicating that some of the considered sequences are more random than other however the method does not give any definite answer to the question of which sequence is more disordered than the other and in this sense the results obtained are negative we compare this with some other ways of measuring the amount of order disorder in such systems and there seems to be no direct correspondence between the measures

spectral and thermodynamic properties of a fibonacci quasicrystal the authors present calculations leading to the phonon density of states in a fibonacci quasicrystal with fixed end points the method is based on a selective elimination process and the results show that the system at low frequencies has the behaviour of a periodic lattice the high frequency end of the spectrum is found to have large forbidden bands and in the limit of longer chains the higher frequency modes are seen to be suppressed the density of state function has further been used to calculate the heat capacity which is found to be lower than its periodic counterpart the electronic energy has been calculated from the associated transmission coefficient and the results show that the spectrum is largely band like in contrast to the uniform scaling structure which has been reported it is also shown that in the limit the quasicrystal reduces to a periodic structure the calculations reproduce the usual results

continuity properties of the electronic spectrum of 1 d quasicrystals in this paper we consider operatorsh x defined onl 2 by h alpha x psi n sum limits m in mathbb z t m circ phi n alpha x psi n m where x x t m is in the algebra of bounded periodic functions on 2 generated by the characteristic functions of the sets phi n left alpha x in mathbb r 2 left left 1 alpha leqq x alpha bmod 1 right right right this class of hamiltonian includes the kohmoto model numerically computed by ostlund and kim where the potential is given by upsilon alpha x n lambda chi 1 alpha 1 x n alpha n in mathbb z x lambda alpha in mathbb r see b i s t we prove that the spectrum as a set ofh x varies continuously with respect to near each irrational for anyx we also show that the various strong limits obtained as converges to a rational numberp q describe either a periodic medium or a periodic medium with a localized impurity the corresponding spectrum has eigenvalues in the gaps and the right and left limits as p q do not coincide for the kohmoto model the results are obtained throughc algebra techniques

trace maps for arbitrary substitution sequences the authors prove the existence and show how to construct trace maps for products of 2 2 matrices generated by arbitrary substitution sequences the dimension of the underlying space of the trace map is lower than the one suggested recently by other authors

spectrum and extended states in a harmonic chain with controlled disorder effects of the thue morse symmetry along the lines of previous work we give the general framework together with a detailed and rigorous study of the spectrum and born von karman eigenstates of a 1d harmonic chain with controlled disorder determined by the thue morse sequence the spectrum is a cantor like set we prove numerically that its measure is zero and calculate its bouligand minkowski dimension box dimension we prove that the value of the ids on each of the gaps is 2k 1 3 2 p withk andp integers finally we also prove that points in a dense subset of the spectrum give rise to extended states an exceptional property due to the symmetry of the thue morse substitution which can have important applications to multilayered structures and we illustrate this situation

anomaly and singular continuous spectrum in a one dimensional incommensurate system the authors study a tight binding model given by psi n 1 psi n 1 v cos 2 pi qn theta psi n e psi n where q is a liouville number it is demonstrated that all eigenstates are critical with singular continuous spectra for all values of v these critical states appear to be composed of connected extended states they also find a band centre anomaly the resistance of the band centre displays coupled oscillations and is much larger than that of other states

power law growth for the resistance in the fibonacci model many one dimensional quasiperiodic systems based on the fibonacci rule such as the tight binding hamiltonianh n n 1 n 1 v n n ne el2 e wherev n n 1 n x denoting the integer part ofx and the golden mean sqrt 5 1 2 give rise to the same recursion relation for the transfer matrices it is proved that the wave functions and the norm of transfer matrices are polynomially bounded critical regime if and only if the energy is in the spectrum of the hamiltonian this solves a conjecture of kohmoto and sutherland on the power law growth of the resistance in a one dimensional quasicrystal

scaling localization and bandwidths for equations with competing periods abstract the finite size scaling theory of the measure of the spectrum of harper s equation is reexamined for sequences of fractions tending to a rational limit a simple criterion is derived which determines whether the corrections to scaling behave as log p 2 or p 2 as the denominator p is increased the question of how special are the properties of the harper equation is studied it is shown that if the pure cosine term in the diagonal term is replaced by a distorted periodic function the different subbands undergo a transition from localized to extended at a value of the strength of the off diagonal term that depends on energy in contrast to the harper equation where the transition is energy independent this has a crucial effect on the measure of the spectrum

a family of generalized fibonacci lattices self similarity and scaling of the wavefunction a family of generalized fibonacci lattices which are locally isomorphic and exhibit a peculiar type of self similarity have been found to exist by shifting the position of the strip initial phase in the projection method the necessary and sufficient conditions of generating such self similar quasiperiodic lattices are established the power law growth behaviour of the wavefunction at e 0 in the off diagonal model defined on some of these lattices has been analysed it is shown that although various structures resulting from different initial phases are in a local isomorphism class they lead to a variety of maximum exponents of power for the scaling of the wavefunction

number of scaling factors in incommensurate systems the authors study the scaling properties of an incommensurate system which obeys simple clustering rules the one dimensional tight binding schrodinger equation with a fibonacci sequence potential they find that the clustering rules do not guarantee a finite number of scaling factors they find the energy spectrum by recursion the clustering rules and recursion relations suggest different descriptions of the same spectrum based on the thermodynamic formalism they compare these descriptions calculate their associated scaling functions and construct the mapping between them

hierarchical band clustering and fractal spectra in incommensurate systems an analytic derivation is given of rules exactly describing the hierarchical band clustering in the spectrum of harper s equation for various incommensurate systems these rules involving transformations of the incommensurability parameter phi are shown to apply exactly for arbitrary coupling strength the approach also provides energy scale factors and hence the fractal dimension df 0 50 or 0 01 of the self similar spectrum occurring when phi takes a fixed point value phi g the golden mean

spectral properties of one dimensional quasi crystals in this paper we prove that the one dimensional schrodinger operator onl 2 with potential given by upsilon n lambda chi 1 alpha 1 x n alpha alpha notin mathbb q has a cantor spectrum of zero lebesgue measure for any irrational and any 0 we can thus extend the kotani result on the absence of absolutely continuous spectrum for this model to all mathtype mtef 2 1 feaafiart1ev1aaatcvaufebsjuyzl2yd9gzlbvynv2caerbulwbln hiov2dgi1btfmbaexatlxbi9gbaerbd9wdylwzybitldharqqtubsr 4rnchbgeagqivu0je9sqqrpepc0xbbl8f4rqqrffpeea0xe9lq jc9 vqaqpepm0xbba9pwe9q8fs0 yqaqpepae9pg0firpepge9fr xfr x frpewzqaaeaabigaciaacaqabeaadaqaaqaaaoqaaiaadihaiiaacq wfiiiztuudjxwak1uy0hmmaehbfv3yslgzg0uy0hgiud3bagqbbiab nj8ubaa 4628 x in mathbb t

anomalous thermostat and intraband discrete breathers abstract we investigate the dynamics of a macroscopic system which consists of an anharmonic subsystem embedded in an arbitrary harmonic lattice including quenched disorder the coupling between both parts is bilinear elimination of the harmonic degrees of freedom leads to a nonlinear langevin equation with memory kernels t and noise term t for the anharmonic coordinates q t q t for zero temperature i e for t 0 we prove that the support of the fourier transform of t and of the time averaged velocity velocity correlation functions k t of the anharmonic system cannot overlap as a consequence the asymptotic solutions can be constant periodic quasiperiodic or almost periodic and possibly weakly chaotic for a sinusoidal trajectory q t with frequency we find that the energy e t transferred to the harmonic system up to time t is proportional to t if equals one of the phonon frequencies it is 2 we prove that there is a zero measure set l such that for in its full measure complement r l it is 0 i e there is no energy dissipation under certain conditions l contains a subset l such that for l the dissipation rate is nonzero and may be subdissipative 0 1 or superdissipative 1 2 compared to ordinary dissipation 1 consequently the harmonic bath does act as an anomalous thermostat in variance with the common belief that elimination of a macroscopically large number of degrees of freedom always generates dissipation forcing convergence to equilibrium intraband discrete breathers are such solutions which do not relax we prove for arbitrary anharmonicity and small but finite coupling that intraband discrete breathers with frequency exist for all in a cantor set c k of finite lebesgue measure this is achieved by estimating the contribution of small denominators appearing for g t related to t for c k the small denominators do not lead to divergencies such that g t is a smooth and bounded function in t

absence of localization in a class of schrodinger operators with quasiperiodic potential we prove that a class of discrete schrodinger operators with a quasiperiodic potential taking only a finite number of values exhibits purely continuous spectrum in particular they cannot have localized eigenvectors

spectral continuity for aperiodic quantum systems i general theory abstract how does the spectrum of a schrodinger operator vary if the corresponding geometry and dynamics change is it possible to define approximations of the spectrum of such operators by defining approximations of the underlying structures in this work a positive answer is provided using the rather general setting of groupoid c algebras a characterization of the convergence of the spectra by the convergence of the underlying structures is proved in order to do so the concept of continuous field of groupoids is slightly extended by adding continuous fields of cocycles with this at hand magnetic schrodinger operators on dynamical systems or delone systems fall into this unified setting various approximations used in computational physics like the periodic or the finite cluster approximations are expressed through the tautological groupoid which provides a universal model for fields of groupoids the use of the hausdorff topology turns out to be fundamental in understanding why and how these approximations work

unbounded trace orbits of thue morse hamiltonian it is well known that an energy is in the spectrum of fibonacci hamiltonian if and only if the corresponding trace orbit is bounded however it is not known whether the same result holds for the thue morse hamiltonian in this paper we give a negative answer to this question more precisely we construct two subsets sigma ii and sigma iii of the spectrum of the thue morse hamiltonian both of which are dense and uncountable such that each energy in sigma ii cup sigma iii corresponds to an unbounded trace orbit exact estimates on the norm of the transfer matrices are also obtained for these energies for e in sigma ii cup sigma iii the norms of the transfer matrices behave like begin aligned e c 1 gamma sqrt n le vert t n e vert le e c 2 gamma sqrt n end aligned however two types of energies are quite different in the sense that each energy in sigma ii is associated with a two sided pseudo localized state while each energy in sigma iii is associated with a one sided pseudo localized state the difference is also reflected by the local dimensions of the spectral measure the local dimension is 0 for energies in sigma ii and is larger than 1 for energies in sigma iii as a comparison we mention another known countable dense subset sigma i each energy in sigma i corresponds to an eventually constant trace map and the associated eigenvector is an extended state in summary the thue morse hamiltonian exhibits mixed spectral nature

a natural class of generalized fibonacci chains in this paper we propose a class of substitution rules that generate quasiperiodic chains sharing their typid properties with the quasiperiodic fibonacci chain for a subclass we explicitly con vct the atomic surface moreover scaling properties of the energy specmm are discussed in relation to the dynamics of trace maps

wave propagation through cantor set media chaos scaling and fractal structures propagation of waves through cantor set media is investigated by renormalization group analysis for specific values of wave numbers transmission coefficients are shown to be governed by the logistic map and in the chaotic region they show sensitive dependence on small changes in parameters of the system such as the index of refraction for other values of wave numbers our numerical results suggest that light transmits completely or reflects completely by the cantor set media text c ensuremath infty it is also shown that transmission coefficients exhibit a local scaling behavior near complete transmission if the complete transmission is achieved at a wave number ensuremath kappa ensuremath kappa ensuremath ast with a rational ensuremath kappa ensuremath ast ensuremath pi the scaling function is obtained analytically by using the euler totient function and the local scaling behavior is confirmed numerically

dynamical properties of quasi crystals fibonacci chain and penrose lattice the discrete scale invariance of quasi periodic systems leads to a scaling relationship omega l zf l between a characteristic dynamic variable omega and length l where f l is a periodic function of ln l the dynamic exponent z can be calculated for diffusion spin wave and phonon dynamics where omega is respectively gamma omega omega 2 where gamma and omega are characteristic rates or frequencies respectively by exploiting a crossover argument which results in z df t where df is the fractal dimension and t is the length scaling exponent for the conductance the latter exponents and hence z are calculated for a fibonacci chain model df t z 1 1 2 and via an exact bond moving technique for the penrose lattice df t z 2 0 2 the technique also provides the thermal exponent of the heisenberg spin model on the two quasi crystals nu 1 infinity respectively

beyond bragg mirrors the design of aperiodic omnidirectional multilayer reflectors a new method that enables omnidirectional reflectivity for all polarizations of incident light over a wide selectable range of wavelengths was used to design aperiodic multilayer mirrors choosing the materials and with the desired threshold value of the reflection coefficient also being given the resulting reflectors work for all polarizations in a predetermined range of incidence angles and wavelengths they are 1d photonic aperiodic crystals composed of a stack of layers arranged according to a deterministic aperiodic substitutive sequence appropriately determined in the calculation of the optical multilayer properties with the rayleigh abeles matrices the use of the computationally efficient building block recurrence or trace antitrace map techniques avoids numerical instability and strongly reduces computational time experimental measurements of a prototype satisfying predetermined requirements fabricated after the sequence generated by a b bba bbba show excellent agreement

wave propagation in quasiperiodic structures stop pass band distribution and prestress effects the band structure of dispersion diagrams for axial and flexural waves of quasiperiodic infinite beams is investigated every structure is composed of a repeated elementary cell generated adopting the fibonacci sequence different terms of this sequence are investigated the problem is then solved evaluating the transmission matrix of the cell and applying the floquet bloch technique in the case of axial vibrations a homogeneous rod where the quasiperiodicity is given by the particular distribution of external springs is considered while for the flexural problem the quasiperiodicity is defined in terms of relative distance between the simple supports that sustain the beam in both cases it is shown that for different fibonacci sequences the number of stop pass bands within a defined range of frequencies changes and follows the fibonacci recursion rule showing a self similar pattern in addition the overall dispersion characteristic can be interpreted in terms of an invariant function of the circular frequency independent of the sequence generating the elementary cell for flexural waves the effects of the axial prestress on dispersion diagrams is highlighted and the frequency shift of the stop pass band positions is quantified it is noted that a tensile axial prestress promotes length reduction of pass bands while leaves almost unchanged the length of stop band intervals

frustration effects in quasicrystals an exactly soluble example in one dimension frustration effects due to competing interactions between magnetic atoms in quasicrystals are studied in an exactly soluble example the ising chain in a quasiperiodic two valued magnetic field at zero temperature the model exhibits an infinity of pure phases characterised by a modulation of the order parameter which reflects the aperiodic structure of the system up to a larger and larger length scale

schrodinger operators with rudin shapiro potentials are not palindromic we prove a conjecture of a hof o knill and b simon commun math phys 174 149 159 1995 by showing that the rudin shapiro sequence is not palindromic i e does not contain arbitrarily long palindromes we prove actually this property for all paperfolding sequences and all rudin shapiro sequences deduced from paperfolding sequences as a consequence and as guessed by the above authors their method cannot be used for establishing that discrete schrodinger operators with rudin shapiro potentials have a purely singular continuous spectrum

escher like quasiperiodic heterostructures quasiperiodic heterostructures present unique structural electronic and vibrational properties connected to the existence of incommensurate periods we go beyond previous schemes such as fibonacci or thue morse based on substitutional sequences by introducing construction rules generated by tessellations of the unit disc by regular polygons we explore some of the properties exhibited by these systems

aperiodic and correlated disorder in xy chains exact results we study thermodynamic properties surface magnetization specific heat and susceptibility of xy quantum chains with coupling constants following arbitrary substitution sequences generalizing an exact renormalization group rg transformation originally formulated for ising quantum chains we obtain exact relevance criteria of harris luck type for this class of models for two letter substitution rules a detailed classification is given of sequences leading to irrelevant marginal or relevant aperiodic modulations we find that the relevance of the same aperiodic sequence of couplings in general will be different for xy and ising quantum chains by our method continuously varying critical exponents may be calculated exactly for arbitrary two letter substitution rules with marginal aperiodicity a number of examples are given including the period doubling three folding and precious mean chains we also discuss extensions of the renormalization approach to a special class of long range correlated random chains generated by random substitutions

dynamical systems for quasiperiodic chains and new self similar polynomials dynamical systems in sl 2 r or sl 2 c naturally appear in the transfer matrix method for quasiperiodic chains characterized by arbitrary irrational numbers we show new subdynamical systems and invariants that are related to full diagonal and off diagonal components of the transfer matrices they are analogous to formulae of chebyshev polynomials of the first and second kinds applying them to an electronic problem on the fibonacci chain we obtain sets of self similar polynomials quasiperiodic extension of the chebyshev polynomials of the first and second kinds with self similar properties two scaling factors of the self similarities coincide with ones obtained by the perturbative decimation renormalization group method

spectral properties of a tight binding hamiltonian with period doubling potential we study a one dimensional tight binding hamiltonian with a potential given by the period doubling sequence we prove that its spectrum is purely singular continuous and supported on a cantor set of zero lebesgue measure for all nonzero values of the potential strength moreover we obtain the exact labelling of all spectral gaps and compute their widths asymptotically for small potential strength

fractal bifurcation sets renormalization strange sets and their universal invariants the transition structure of the most common routes to chaos are organized by fractal bifurcation sets examples include the quasiperiodic transitions to chaos and the period doubling structure found in arnol d tongues in this paper i discuss the universality of such fractal bifurcation sets and their relation to strange invariant sets of renormalization transformations an important result is that fractal bifurcation sets from within the same universality chaos are lipeomorphic this implies that they have the same fractal structure and in particular the same hausdorff dimension and scaling spectra some other invariants are introduced

almost periodic one dimensional systems the schrodinger equation with potential v x 2 1cos x 2 2cos x is considered its solution reduces to the problem of finding the eigenvectors of a matrix the eigenvalues of this matrix show a band structure which is very sensitive to the value of the parameter the solutions of the schrodinger equation are presented and their physical meaning is discussed the potential v x has a complex multiwell structure and quantum tunneling occurs the accuracy of all approximations is carefully studied

repeller structure in a hierarchical model i topological properties the repeller associated with the renormalization dynamics of the spectral problem of a hierarchical tight binding schrodinger equation is studied analysis of escaping regions and of stable and unstable manifolds provide complementary descriptions of the recurrent set whose structure undergoes relevant changes when the growth rater of the potential barriers is modified the minimal region containing the repeller is determined and the mechanism originating a cantor set structure along the unstable manifold is revealed the repeller is continuous along the stable manifold forr 2 finally we show the existence of a pointlike component of the spectrum located at its upper extremum forr 1 and we present the associated wavefunctions

electronic stability of disordered systems application to quasicrystals the author studies energetic stability of disordered binary alloys in which atoms are interacting via a tight binding hamiltonian in 1d for a weak potential the gain in energy compared to the periodic linear chain is derived as a function of the electronic filling factor and the disorder very precise results are given in the case of a quasiperiodic arrangement this kind of phase plays a very particular role since it is shown that for a given filling factor v or fermi level position the most stable structure is a quasiperiodic one associated to v moreover for this structure the fluctuations of the density are unbounded if v and the stoichiometric coefficients do not fulfil an arithmetical condition different features are observed for rational and irrational values of v

singular continuous electron spectrum for a class of circle sequences we derive substitution rules for a class of binary quasiperiodic sequences generated by circle maps whose rotation numbers are obtained from the precious means the nature of the electron spectra for the corresponding diatomic chains is studied in the nearest neighbour on site tight binding approximation using the transfer matrix technique by studying properties of the trace maps we find that the spectrum is purely singular continuous in most of the studied cases

cellular automata and statistical mechanical models we elaborate on the analogy between the transfer matrix of usual lattice models and the master equation describing the time development of cellular automata transient and stationary properties of probabilistic automata are linked to surface and bulk properties respectively of restricted statistical mechanical systems it is demonstrated that methods of statistical physics can be successfully used to describe the dynamic and the stationary behavior of such automata some exact results are derived including duality transformations exact mappings disorder and linear solutions many examples are worked out in detail to demonstrate how to use statistical physics in order to construct cellular automata with desired properties this approach is considered to be a first step toward the design of fully parallel probabilistic systems whose computational abilities rely on the cooperative behavior of their components

structure factor of 1d systems superlattices based on two letter substitution rules i delta bragg peaks the recent generalization to the case of arbitrary tile lengths and arbitrary scattering factors of the calculation of the structure factor of 1d substitutional systems is studied in detail this method makes a easy to find all the peaks in the diffraction spectrum of a system the well known periodic and quasiperiodic spectra with delta peaks at integer multiples of a single number and integer linear combinations of two incommensurate frequencies respectively were found to be the l 0 subsets of two more general types of spectra infinite periodic or limit periodic and infinite quasiperiodic or limit quasiperiodic characterized by rational numbers of the type m nl l 0 infinity in place of the above integers substitution rules that produce quasicrystalline quasiperiodic and infinite quasiperiodic spectra give the same type of spectrum for all values of the ratio rho rho a rho b of the two tile lengths rho a and rho b this is not the case for the other rules thus the same substitution rule such as the copper mean rule can give an infinite periodic spectrum for a single rational ratio rho rho a rho b of the two tile lengths rho a and rho b a periodic like spectrum for other rational rho and a spectrum in some aspects similar to that of a random system when rho is an irrational number on the other hand a thue morse system diffracts as a periodic crystal when p not 1 but has no non trivial delta peaks when rho 1 other thue morse like systems can have infinite periodic spectra for all rho

resistance of one dimensional quasicrystals abstract one dimensional quasiperiodic systems are often based on an inflation rule giving rise to a recursion formula on the transfer matrices it is shown in a general setting that the norm growth of these matrices can be evaluated a consequence is for instance a purely critical regime namely polynomial bounds in function of the sample size for the electrical resistance when the energy is in the spectrum this solves a conjecture by sutherland and kohmoto