determinants on von neumann algebras mahler measures and ljapunov exponents for an ergodic automorphism of a probability space a p consider the semidirect product von neumann algebra r l z let u be the unitary operator corresponding to the action of then the finite sums of the form i aiu i with ai in l form a strongly dense subalgebra of r in this paper we calculate the fuglede kadison determinant detr for a certain class of elements the answer is given in terms of the ljapunov exponents i of a matrix a over the ring l which is attached to by a simple receipe the formula has the form

non stationary correlation matrices and noise abstract the exact meaning of the noise spectrum of eigenvalues of the correlation matrix is discussed in order to better understand the possible phenomena behind the observed noise the spectrum of eigenvalues of the correlation matrix is studied under a model where most of the true eigenvalues are zero and the parameters are non stationary the results are compared with real observation of brazilian assets suggesting that although the non stationarity seems to be an important aspect of the problem partially explaining some of the eigenvalues as well as part of the kurtosis of the assets it cannot by itself provide all the corrections needed to make the proposed model fit the data perfectly

estimated correlation matrices and portfolio optimization abstract correlations of returns on various assets play a central role in financial theory and also in many practical applications from a theoretical point of view the main interest lies in the proper description of the structure and dynamics of correlations whereas for the practitioner the emphasis is on the ability of the models to provide adequate inputs for the numerous portfolio and risk management procedures used in the financial industry the theory of portfolios initiated by markowitz has suffered from the curse of dimensions from the very outset over the past decades a large number of different techniques have been developed to tackle this problem and reduce the effective dimension of large bank portfolios but the efficiency and reliability of these procedures are extremely hard to assess or compare in this paper we propose a model simulation based approach which can be used for the systematical testing of all these dimensional reduction techniques to illustrate the usefulness of our framework we develop several toy models that display some of the main characteristic features of empirical correlations and generate artificial time series from them then we regard these time series as empirical data and reconstruct the corresponding correlation matrices which will inevitably contain a certain amount of noise due to the finiteness of the time series next we apply several correlation matrix estimators and dimension reduction techniques introduced in the literature and or applied in practice as in our artificial world the only source of error is the finite length of the time series and in addition the true model hence also the true correlation matrix are precisely known therefore in sharp contrast with empirical studies we can precisely compare the performance of the various noise reduction techniques one of our recurrent observations is that the recently introduced filtering technique based on random matrix theory performs consistently well in all the investigated cases based on this experience we believe that our simulation based approach can also be useful for the systematic investigation of several related problems of current interest in finance

higher order properties of gmm and generalized empirical likelihood estimators in an effort to improve the small sample properties of generalized method of moments gmm estimators a number of alternative estimators have been suggested these include empirical likelihood el continuous updating and exponential tilting estimators we show that these estimators share a common structure being members of a class of generalized empirical likelihood gel estimators we use this structure to compare their higher order asymptotic properties we find that gel has no asymptotic bias due to correlation of the moment functions with their jacobian eliminating an important source of bias for gmm in models with endogeneity we also find that el has no asymptotic bias from estimating the optimal weight matrix eliminating a further important source of bias for gmm in panel data models we give bias corrected gmm and gel estimators we also show that bias corrected el inherits the higher order property of maximum likelihood that it is higher order asymptotically efficient relative to the other bias corrected estimators

dynamics of competition between collectivity and noise in the stock market abstract detailed study of the financial empirical correlation matrix of the 30 companies which deutsche aktienindex dax comprised during the period of the last 11 years using the time window of 30 trading days is presented this allows clear identification of a nontrivial time dependence of the resulting correlations in addition as a rule the drawdowns are always accompanied by a sizable separation of one strong collective eigenstate of the correlation matrix which at the same time reduces the variance of the noise states the opposite applies to drawups in this case the dynamics spreads more uniformly over the eigenstates which results in an increase of the total information entropy analogous study of the market corresponding to daw jones industrial average djia leads to similar conclusions in the latter case however the correlations are weaker on average one possible reason for this effect is that the market represented by djia is less susceptible to various external factors than the one represented by dax

mixing automorphisms of compact groups and a theorem of schlickewei we prove that every mixing d by automorphisms of a compact connected abelian group is mixing of all orders

spectral analysis of time dependent market adjusted return correlation matrix abstract we present an adjusted method for calculating the eigenvalues of a time dependent return correlation matrix in a moving window first we compare the normalized maximum eigenvalue time series of the market adjusted return correlation matrix to that of the logarithmic return correlation matrix on an 18 year dataset of 310 s p 500 listed stocks for small and large window or memory sizes we observe that the resulting new eigenvalue time series is more stationary than the time series obtained without the adjustment second we perform this analysis while sweeping the window size 5 100 500 in order to examine the dependence on the choice of window size this approach demonstrates the multi modality of the eigenvalue distributions we find that the three dimensional distribution of the eigenvalue time series for our market adjusted return is significantly more stationary than that produced by classic method finally our model offers an approximate polarization domain characterized by a smooth l shaped strip the polarization with large amplitude is revealed while there is persistence in agreement of individual stock returns movement with market with small amplitude most of the time

synchronization points and associated dynamical invariants this paper introduces new invariants for multiparameter dynamical systems this is done by counting the number of points whose orbits intersect at time n under simultaneous iteration of finitely many endomorphisms we call these points synchronization points the resulting sequences of counts together with generating functions and growth rates are subsequently investigated for homeomorphisms of compact metric spaces toral automorphisms and compact abelian group epimorphisms synchronization points are also used to generate invariant measures and the distribution properties of these are analysed for the algebraic systems considered furthermore these systems reveal strong connections between the new invariants and problems of active interest in number theory relating to heights and greatest common divisors

noise dressing of financial correlation matrices we show that results from the theory of random matrices are potentially of great interest to understand the statistical structure of the empirical correlation matrices appearing in the study of multivariate time series the central result of the present study which focuses on the case of financial price fluctuations is the remarkable agreement between the theoretical prediction based on the assumption that the correlation matrix is random and empirical data concerning the density of eigenvalues associated to the time series of the different stocks of the s 500 or other major markets in particular the present study raises serious doubts on the blind use of empirical correlation matrices for risk management

multifractality in the random parameter model for multivariate time series abstract the random parameter model was proposed to explain the structure of the covariance matrix in problems where most but not all of the eigenvalues of the covariance matrix can be explained by random matrix theory in this article we explore the scaling properties of the model as observed in the multifractal structure of the simulated time series we use the wavelet transform modulus maxima technique to obtain the multifractal spectrum dependence with the parameters of the model the model shows a scaling structure compatible with the stylized facts for a reasonable choice of the parameter values

the bulk of the stock market correlation matrix is not pure noise abstract we analyse the structure of the distribution of eigenvalues of the stock market correlation matrix with increasing length of the time series representing the price changes we use 100 highly capitalized stocks from the american market and relate the result to the corresponding ensemble of wishart random matrices it turns out that systematically more eigenvalues remain beyond the borders prescribed by this variant of the random matrix theory rmt this may indicate that even the bulk of the spectrum of the stock market correlation matrix carries some sort of correlations that are masked by a measurement noise when the time series used to construct the matrix are short we also study some other characteristics of the noisy eigensignals like their return distributions temporal correlations or their multifractal spectra and the results support the above conclusions

entropy of automorphisms of ii1 factors arising from the dynamical systems theory abstract let a countable amenable group g act freely and ergodically on a lebesgue space x preserving the measure if t aut x is an automorphism of the equivalence relation defined by g then t can be extended to an automorphism t of the ii 1 factor m l x g we prove that if t commutes with the action of g then h t h t where h t is the connes stormer entropy of t and h t is the kolmogorov sinai entropy of t we also prove that for given s and t 0 s t there exists a t such that h t s and h t t

random matrix approach to cross correlations in financial data we analyze cross correlations between price fluctuations of different stocks using methods of random matrix theory rmt using two large databases we calculate cross correlation matrices c of returns constructed from i 30 min returns of 1000 us stocks for the 2 yr period 1994 1995 ii 30 min returns of 881 us stocks for the 2 yr period 1996 1997 and iii 1 day returns of 422 us stocks for the 35 yr period 1962 1996 we test the statistics of the eigenvalues lambda i of c against a null hypothesis a random correlation matrix constructed from mutually uncorrelated time series we find that a majority of the eigenvalues of c fall within the rmt bounds lambda lambda for the eigenvalues of random correlation matrices we test the eigenvalues of c within the rmt bound for universal properties of random matrices and find good agreement with the results for the gaussian orthogonal ensemble of random matrices implying a large degree of randomness in the measured cross correlation coefficients further we find that the distribution of eigenvector components for the eigenvectors corresponding to the eigenvalues outside the rmt bound display systematic deviations from the rmt prediction in addition we find that these deviating eigenvectors are stable in time we analyze the components of the deviating eigenvectors and find that the largest eigenvalue corresponds to an influence common to all stocks our analysis of the remaining deviating eigenvectors shows distinct groups whose identities correspond to conventionally identified business sectors finally we discuss applications to the construction of portfolios of stocks that have a stable ratio of risk to return less

isomorphism rigidity of irreducible algebraic ℤd actions an irreducible algebraic d action on a compact abelian group x is a d action by automorphisms of x such that every closed invariant subgroup y x is finite we prove the following result if d 2 then every measurable conjugacy between irreducible and mixing algebraic d actions on compact zero dimensional abelian groups is affine for irreducible expansive and mixing algebraic d actions on compact connected abelian groups the analogous statement follows essentially from a result by katok and spatzier on invariant measures of such actions cf 4 and 3 by combining these two theorems one obtains isomorphism rigidity of all irreducible expansive and mixing algebraic d actions with d 2

random but not so much a parameterization for the returns and correlation matrix of financial time series abstract a parameterization that is a modified version of a previous work is proposed for the returns and correlation matrix of financial time series and its properties are studied this parameterization allows easy introduction of non stationarity and it shows several of the characteristics of the true observed realizations such as fat tails volatility clustering and a spectrum of eigenvalues of the correlation matrix that can be understood as an extension of random matrix theory results the predicted behavior of this parameterization for the eigenvalues is compared with the eigenvalues of brazilian assets and it is shown that those predictions fit the data better than random matrix theory

abelian sandpiles and the harmonic model we present a construction of an entropy preserving equivariant surjective map from the d dimensional critical sandpile model to a certain closed shift invariant subgroup of mathbb t mathbb z d the harmonic model a similar map is constructed for the dissipative abelian sandpile model and is used to prove uniqueness and the bernoulli property of the measure of maximal entropy for that model

mahler measure and entropy for commuting automorphisms of compact groups we compute the joint entropy ofd commuting automorphisms of a compact metrizable group letr d u 1 1 d 1 1 be the ring of laurent polynomials ind commuting variables andm be anr d module then the dual groupx m ofm is compact and multiplication onm by each of thed variables corresponds to an action m of d by automorphisms ofx m every action of d by automorphisms of a compact abelian group arises this way iff r d our main formula shows that the topological entropy of alpha r d left langle f right rangle is given by h left alpha r d left langle f right rangle right log m left f right int limits 0 1 ldots int limits 0 1 log left f left e 2 pi it 1 ldots e 2 pi it d right right dt 1 ldots dt d where m f is the mahler measure off this reduces to the classical result for toral automorphisms via jensen s formula while the entropy of a single automorphism of a compact group is always the logarithm of an algebraic integer this no longer seems to hold for joint entropy of commuting automorphisms since values such as 7 3 4 2 occur if p is a non principal prime ideal we show h left alpha r d p right 0 using an analogue of the yuzvinskii thomas addition formula we computeh m for arbitraryr d modulesm and then the joint entropy for an action of d on a not necessarily abelian compact group using a result of boyd we characterize those m which have completely positive entropy in terms of the prime ideals associated tom and show this condition implies that m is mixing of all orders we also establish an analogue of berg s theorem proving that if m has finite entropy then haar measure is the unique measure of maximal entropy if and only if m has completely positive entropy finally we show that for expansive actions the growth rate of the number of periodic points equals the topological entropy

algebraic ℤ actions of entropy rank one we investigate algebraic z d actions of entropy rank one namely those for which each element has finite entropy such actions can be completely described in terms of diagonal actions on products of local fields using standard adelic machinery this leads to numerous alternative characterizations of entropy rank one both geometric and algebraic we then compute the measure entropy of a class of skew products where the fiber maps are elements from an algebraic z d action of entropy rank one this leads via the relative variational principle to a formula for the topological entropy of continuous skew products as the maximum of a finite number of topological pressures we use this to settle a conjecture concerning the relational entropy of commuting toral automorphisms

bowen topological entropy of subsets for amenable group actions abstract we extend the definition of bowen topological entropy of subsets to continuous action of amenable groups on a compact metrizable space we investigate how bowen topological entropy behaves for restricted actions of finite index subgroups and verify the validity of the subgroup formula we also show that the bowen topological entropy of subsets for amenable group actions can be determined via the local entropies of measures as applications we compute the bowen topological entropy of a subset for bernoulli shift over amenable groups

isomorphism rigidity of commuting automorphisms r let d 1 and let x and y be two zero entropy z d actions on compact abelian groups by d commuting automorphisms we show that if all lower rank subactions of and have completely positive entropy then any measurable equivariant map from x to y is an affine map in particular two such actions are measurably conjugate if and only if they are algebraically conjugate

measures of globalization based on cross correlations of world financial indices abstract the cross correlation matrix of daily returns of stock market indices in a diverse set of 37 countries worldwide was analyzed comparison of the spectrum of this matrix with predictions of random matrix theory provides an empirical evidence of strong interactions between individual economies as manifested by three largest eigenvalues and the corresponding set of stable non random eigenvectors the observed correlation structure is robust with respect to changes in the time horizon of returns ranging from 1 to 10 trading days and to replacing individual returns with just their signs this last observation confirms that it is correlations between signs and not absolute values of fluctuations which are mostly responsible for the observed effect negative changes in the index are somewhat more correlated than the positive ones also in our data set the reaction of asian stock indices to changes in european and american ones persists for about 3 days

entropy and the variational principle for actions of sofic groups recently lewis bowen introduced a notion of entropy for measure preserving actions of a countable sofic group on a standard probability space admitting a generating partition with finite entropy by applying an operator algebra perspective we develop a more general approach to sofic entropy which produces both measure and topological dynamical invariants and we establish the variational principle in this context in the case of residually finite groups we use the variational principle to compute the topological entropy of principal algebraic actions whose defining group ring element is invertible in the full group c algebra

a variational principle for free semigroup actions abstract in this paper we introduce a notion of measure theoretical entropy for a finitely generated free semigroup action and establish a variational principle when the semigroup is generated by continuous self maps on a compact metric space and has finite topological entropy in the case of semigroups generated by ruelle expanding maps we prove the existence of equilibrium states and describe some of their properties of independent interest are the different ways we will present to compute the metric entropy and a characterization of the stationary measures

almost block independence and bernoullicity of ℤ d actions by automorphisms of compact abelian groups we prove that a d action by automorphisms of a compact abelian group is bernoulli if and only if it has completely positive entropy the key ingredients of the proof are the extension of certain notions of asymptotic block independence from actions to d action and their equivalence with bernoullicity and a surprisingly close link between one of these asymptotic block independence properties for d actions by automorphisms of compact abelian groups and the product formula for valuations on global fields

group automorphisms with prescribed growth of periodic points and small primes in arithmetic progressions in intervals abstract we investigate the question of which growth rates are possible for the number of periodic points of a compact group automorphism our arguments involve a modification of linnik s theorem concerning small prime numbers in arithmetic progressions which lie in intervals

fuglede kadison determinants and entropy for actions of discrete amenable groups consider a discrete group t and an element in the integral group ring zt then t acts from the left on the discrete additive group zt ztf by automorphisms of groups dualizing we obtain a left t action on the compact pontrjagin dual group xf zt ztf by continuous automorphisms of groups by definition xf is a closed subshift of s1 1 for measurable or topological actions of amenable groups a good entropy theory is available ow m thus from now on let t be a finitely generated discrete amenable group we are interested in determining the topological entropy hf of the t action on xf it agrees with the measure theoretic entropy of the t action on xf equipped with its haar probability measure in their paper fk from 1952 fuglede and kadison introduced a determinant for units in finite factor von neumann algebras more recently l ck has general ized their theory to nonunits and to group von neumann algebras which are not necessarily factors l2 he needed this to define l2 torsion for general covering spaces the von neumann algebra a t of the amenable group t is a certain completion of cr and hence we may consider the fuglede kadison l ck determinant detjvr of we refer to section 6 for the definition of a log strong f0lner sequence it is known that finitely generated virtually nilpotent groups have log strong f0lner sequences our main result is

signal and noise in correlation matrix using random matrix technique we determine an exact relation between the eigenvalue spectrum of the covariance matrix and of its estimator this relation can be used in practice to compute eigenvalue invariants of the covariance correlation matrix results can be applied in various problems where one experimentally estimates correlations in a system with many degrees of freedom like for instance those in statistical physics lattice measurements of field theory genetics quantitative finance and other applications of multivariate statistics

information and entropy econometrics editor s view amos golan department of economics american university roper 200 4400 massachusetts ave nw washington dc 20016 usa

universal and nonuniversal properties of cross correlations in financial time series we use methods of random matrix theory to analyze the cross correlation matrix c of stock price changes of the largest 1000 u s companies for the 2 year period 1994 1995 we find that the statistics of most of the eigenvalues in the spectrum of c agree with the predictions of random matrix theory but there are deviations for a few of the largest eigenvalues we find that c has the universal properties of the gaussian orthogonal ensemble of random matrices furthermore we analyze the eigenvectors of c through their inverse participation ratio and find eigenvectors with large ratios at both edges of the eigenvalue spectrum a situation reminiscent of localization theory results

deligne periods of mixed motives theory and the entropy of certain ℤⁿ actions for a coherent sheaf m on the split n torus gn a spec a 7n over a commutative ring a there is a natural en action on its group of global sections f m f g aa m if we give f m the discrete topology we obtain a en action on the compact abelian pontrjagin dual f m the dynamical properties of this action for a z have been investigated for some time especially by k schmidt his book sch gives a comprehensive account of the theory developed so far in particular one may ask for the entropy 0 h m oc of the en action on f m i which measures to what extent repeated application of the action scatters around points note that for any action of zn on a compact abelian group topological entropy and metric entropy with respect to haar measure coincide the results of lind schmidt and ward lsw see also sch chap v can be formulated as follows a z the entropy of m is finite if and only if m belongs to the category t of coherent torsion sheaves on gn by yuzvinskii s addition formula sch th 14 1 entropy defines a homomorphism from the grothendieck group of t to the reals h ko t ri so that it suffices to calculate h oz for irreducible closed subschemes z c n with z g n if z is not defined by a laurent polynomial then h oz 0 on the other hand for any

dimension and entropy in compact topological groups abstract we study the topological entropy h f of continuous endomorphisms f of compact like groups more specifically we consider the e spectrum e t o p k for a compact like group k namely the set of all values h f when f runs over the set e n d k of all continuous endomorphisms of k we pay particular attention to the class e of topological groups without continuous endomorphisms of infinite entropy i e e t o p k as well as the subclass e 0 of e consisting of those groups k with e t o p k 0 it turns out that the properties of the e spectrum and these two classes are very closely related to the topological dimension we show among others that a compact connected group k with finite dimensional commutator subgroup belongs to e if and only if dim k and we obtain a simple formula involving the entropy function for the dimension of an abelian topological group which is either locally compact or bounded in particular compact examples are provided to show the necessity of the compactness or commutativity conditions imposed for the validity of these results e g compact connected semi simple groups k with dim k and k e 0 or countably compact connected abelian groups with the same property since the class e is not stable under taking closed subgroups or quotients we study also the largest subclasses s e and q e respectively of e having these stability properties we provide a complete description of these two classes in the case of compact groups that are either abelian or connected the counterpart for s e 0 and q e 0 is done as well

compact group automorphisms addition formulas and fuglede kadison determinants for a countable amenable group and an element f in the integral group ring z being invertible in the group von neumann algebra of we show that the entropy of the shift action of on the pontryagin dual of the quotient of z by its left ideal generated by f is the logarithm of the fuglede kadison determinant of f for the proof we establish an p version of rufus bowen s denition of topological entropy addition formulas for group extensions of countable amenable group actions and an approximation formula for the fuglede kadison determinant of f in terms of the determinants of perturbations of the compressions of f

signal and noise in financial correlation matrices abstract using random matrix theory one can derive exact relations between the eigenvalue spectrum of the covariance matrix and the eigenvalue spectrum of its estimator experimentally measured correlation matrix these relations will be used to analyze a particular case of the correlations in financial series and to show that contrary to earlier claims correlations can be measured also in the random part of the spectrum implications for the portfolio optimization are briefly discussed

local characteristics entropy and limit theorems for spanning trees and domino tilings via transfer impedances let g be a finite graph or an infinite graph on which mathbb z d acts with finite fundamental domain if g is finite let mathbf t be a random spanning tree chosen uniformly from all spanning trees of g if g is infinite methods from pemantle show that this still makes sense producing a random essential spanning forest of g a method for calculating local characteristics i e finite dimensional marginals of mathbf t from the transfer impedance matrix is presented this differs from the classical matrix tree theorem in that only small pieces of the matrix n dimensional minors are needed to compute small n dimensional marginals calculation of the matrix entries relies on the calculation of the green s function for g which is not a local calculation however it is shown how the calculation of the green s function may be reduced to a finite computation in the case when g is an infinite graph admitting a z d action with finite quotient the same computation also gives the entropy of the law of mathbf t these results are applied to the problem of tiling certain lattices by dominos the so called dimer problem another application of these results is to prove modified versions of conjectures of aldous on the limiting distribution of degrees of a vertex and on the local structure near a vertex of a uniform random spanning tree in a lattice whose dimension is going to infinity included is a generalization of moments to tree valued random variables and criteria for these generalized moments to determine a distribution

theory coherent forecasting we consider a method for producing multivariate density forecasts that satisfy moment restrictions implied by economic theory such as euler conditions the method starts from a base forecast that might not satisfy the theoretical restrictions and forces it to satisfy the moment conditions using exponential tilting although exponential tilting has been considered before in a bayesian context robertson et al 2005 our main contributions are 1 to adapt the method to a classical inferential context with out of sample evaluation objectives and parameter estimation uncertainty and 2 to formally discuss the conditions under which the method delivers improvements in forecast accuracy an empirical illustration which incorporates euler conditions into forecasts produced by bayesian vector autoregressions shows that the improvements in accuracy can be sizable and significant

mean dimension mean rank and von neumann luck rank we introduce an invariant called mean rank for any module m of the integral group ring of a discrete amenable group as an analogue of the rank of an abelian group it is shown that the mean dimension of the induced action on the pontryagin dual of m the mean rank of m and the von neumann luck rank of m all coincide as applications we establish an addition formula for mean dimension of algebraic actions prove the analogue of the pontryagin schnirelmnn theorem for algebraic actions and show that for elementary amenable groups with an upper bound on the orders of finite subgroups algebraic actions with zero mean dimension are inverse limits of finite entropy actions

algebraic entropy for amenable semigroup actions abstract we introduce two notions of algebraic entropy for actions of cancellative right amenable semigroups s on discrete abelian groups a by endomorphisms they extend the classical algebraic entropy for endomorphisms of abelian groups corresponding to the case s n we investigate the fundamental properties of the algebraic entropy and compute it in several examples paying special attention to the case when s is an amenable group for actions of cancellative right amenable monoids on torsion abelian groups we prove the so called addition theorem in the same setting we see that a bridge theorem connects the algebraic entropy with the topological entropy of the dual action by means of pontryagin duality so that we derive an addition theorem for the topological entropy of actions of cancellative left amenable monoids on totally disconnected compact abelian groups

an information theoretic alternative to generalized method of moments estimation while optimally weighted gmm estimation has desirable large sample properties its small sample performance is poor in some applications we propose a computationally simple alternative for weakly dependent data generating mechanisms based on minimization of the kullback leibler information criterion conditions are derived under which the large sample properties of this estimator are similar to gmm i e the estimator will be consistent and asymptotically normal with the same asymptotic covariance matrix as gmm in addition we propose overidentifying and parametric restrictions tests as alternatives to analogous gmm procedures

a measure conjugacy invariant for free group actions this paper introduces a new measure conjugacy invariant for actions of free groups using this invariant it is shown that two bernoulli shifts over a finitely generated free group are measurably conjugate if and only if their base measures have the same entropy this answers a question of ornstein and weiss

some remarks on the entropy for algebraic actions of amenable groups in this short note we study the entropy for algebraic actions of certain amenable groups the possible values for this entropy are studied various fundamental results about certain classes of amenable groups are reproved using elementary arguments and the entropy invariant we provide a natural decomposition of the entropy into summands contributed by individual primes and a summand corresponding to these results extend previous work by lind and ward on p adic entropy

an lp version of von neumann dimension for banach space representations of sofic groups abstract in 5 a gournay defined a notion of l p dimension for invariant subspaces of l q n with amenable the number dim l q l p v is dim v when p q and is preserved by a certain class of equivariant bounded linear isomorphisms we develop a notion of dim l p y where y is a banach space with a uniformly bounded action of a sofic group and is a sofic approximation in particular our definition makes sense for a large class of non amenable groups we also develop a notion of dim s p y with an r embeddable group and s p the space of finite dimensional schatten p class operators these numbers are invariant under bounded equivariant linear isomorphisms and under the natural translation action of dim l p l p v dim v and dim s p l p v dim v for 1 p 2 in particular this shows that l p v is not isomorphic to l p w as a representation of if dim v dim w and is r embeddable we discuss other concrete computations in a follow up paper including proving that our dimension agrees with von neumann dimension for representations contained in a multiple of the left regular representation

choosing a spanning tree for the integer lattice uniformly consider the nearest neighbor graph for the integer lattice zd in d dimensions for a large finite piece of it consider choosing a spanning tree for that piece uniformly among all possible subgraphs that are spanning trees as the piece gets larger this approaches a limiting measure on the set of spanning graphs for zd this is shown to be a tree if and only if d 4 in this case the tree has only one topological end that is there are no doubly infinite paths when d 2 5 the spanning forest has infinitely many components almost surely with each component having one or two topological ends

sofic mean dimension abstract we introduce mean dimensions for continuous actions of countable sofic groups on compact metrizable spaces these generalize the gromov lindenstrauss weiss mean dimensions for actions of countable amenable groups and are useful for distinguishing continuous actions of countable sofic groups with infinite entropy

homoclinic points of algebraic ℤ actions an algebraic zd action is an action of zd by continuous automorphisms of a compact abelian group the dynamics of a single group automorphism have been investigated in great detail over the past several decades more recently the study of algebraic zd actions for d 2 has revealed a striking interplay between these actions and commutative algebra in 2 we summarize those parts of this interaction needed here the purpose of this paper is to study the homoclinic points of algebraic zd actions let al be an algebraic zd action on the compact abelian group x and let ox denote the additive identity of x a point x e x is homoclinic for ce if c nx ox as llnll oo the set ai x of all homoclinic points for oc is clearly a subgroup of x which we call the homoclinic group of oz in 3 we discuss some elementary properties of the homoclinic group including counltability of za x whenever oz is expansive our two main results are contained in 4 these are that if oz is an expansive algebraic ed action then 1 a x is nontrivial if and only if oz has strictly positive entropy and 2 a x is nontrivial and dense in x if and only if oz has completely positive entropy the second result is proved by first establishing in lemma 4 5 the density of a x for certain principal expansive actions by use of fourier analysis for these actions a x is generated by a single fundamental homoclinic point which can be computed explicitly this lemma is then combined with some commutative algebra to prove 2 recent work of kaminker and putnam 3 11 has suggested a general duality in the k theory of c algebras for a principal expansive action al on x we show that a x is isomorphic to the dual group of x providing a class of examples to which their duality theory applies ruelle has investigated expansive topological zd actions which satisfy an orbit tracing property called specification see 12 and 13 showing that there is a thermodynamic formalism for such actions in 5 we show that expansive algebraic zd actions with completely positive entropy always satisfy very strong specification properties thereby providing an extensive class of examples to which the thermodynamic formalism applies

uniform spanning forests we study uniform spanning forest measures on infinite graphs which are weak limits of uniform spanning tree measures from finite subgraphs these limits can be taken with free fsf or wired wsf boundary conditions pemantle proved that the free and wired spanning forests coincide in z d and that they give a single tree iff d 4 in the present work we extend pemantle s alternative to general graphs and exhibit further connections of uniform spanning forests to random walks potential theory invariant percolation and amenability the uniform spanning forest model is related to random cluster models in statistical physics but because of the preceding connections its analysis can be carried further among our results are the following the fsf and wsf in a graph g coincide iff all harmonic dirichlet functions on g are constant the tail fields of the wsf and the fsf are trivial on any graph on any cayley graph that is not a finite extension of z all component trees of the wsf have one end this is new in z d for d 5 on any tree as well as on any graph with spectral radius less than 1 a s all components of the wsf are recurrent the basic topology of the free and the wired uniform spanning forest measures on lattices in hyperbolic space h d is analyzed a cayley graph is amenable iff for all e 0 the union of the wsf and bernoulli percolation with parameter e is connected harmonic measure from infinity is shown to exist on any recurrent proper planar graph with finite codegrees we also present numerous open problems and conjectures

hyperlinearity essentially free actions and l2 invariants the sofic property we prove that connes embedding conjecture holds for the von neumann algebras of sofic groups that is sofic groups are hyperlinear hence we provide some new examples of hyperlinearity we also show that the determinant conjecture holds for sofic groups as well we introduce the notion of essentially free actions and amenable actions and study their properties

harmonic models and spanning forests of residually finite groups abstract we prove a number of identities relating the sofic entropy of a certain class of non expansive algebraic dynamical systems the sofic entropy of the wired spanning forest and the tree entropy of cayley graphs of residually finite groups we also show that homoclinic points and periodic points in harmonic models are dense under general conditions

collective origin of the coexistence of apparent random matrix theory noise and of factors in large sample correlation matrices abstract through simple analytical calculations and numerical simulations we demonstrate the generic existence of a self organized macroscopic state in any large multivariate system possessing non vanishing average correlations between a finite fraction of all pairs of elements the coexistence of an eigenvalue spectrum predicted by random matrix theory rmt and a few very large eigenvalues in large empirical correlation matrices is shown to result from a bottom up collective effect of the underlying time series rather than a top down impact of factors our results in excellent agreement with previous results obtained on large financial correlation matrices show that there is relevant information also in the bulk of the eigenvalue spectrum and rationalize the presence of market factors previously introduced in an ad hoc manner

random matrix theory analysis of cross correlations in financial markets we confirm universal behaviors such as eigenvalue distribution and spacings predicted by random matrix theory rmt for the cross correlation matrix of the daily stock prices of tokyo stock exchange from 1993 to 2001 which have been reported for new york stock exchange in previous studies it is shown that the random part of the eigenvalue distribution of the cross correlation matrix is stable even when deterministic correlations are present some deviations in the small eigenvalue statistics outside the bounds of the universality class of rmt are not completely explained with the deterministic correlations as proposed in previous studies we study the effect of randomness on deterministic correlations and find that randomness causes a repulsion between deterministic eigenvalues and the random eigenvalues this is interpreted as a reminiscent of level repulsion in rmt and explains some deviations from the previous studies observed in the market data we also study correlated groups of issues in these markets and propose a refined method to identify correlated groups based on rmt some characteristic differences between properties of tokyo stock exchange and new york stock exchange are found

entropy determinants and l 2 torsion we show that for any amenable group and any z module m of type fl with vanishing euler characteristic the entropy of the natural action on the pontryagin dual of m is equal to the l 2 torsion of m as a particular case the entropy of the principal algebraic action associated with the module z z f is equal to the logarithm of the fuglede kadison determinant of f whenever f is a non zero divisor in z this conrms a conjecture of deninger as a key step in the proof we provide a general szeg o type approximation theorem for the fuglede kadison determinant on the group von neumann algebra of an amenable group as a consequence of the equality between l 2 torsion and entropy we show that the l 2 torsion of a non trivial amenable group with nite classifying space vanishes this was conjectured by luck finally we establish a milnor turaev formula for the l 2 torsion of a nite acyclic chain complex

sofic mean length abstract given a length function l on the r modules of a unital ring r for each sofic group we define a mean length for every locally l finite r module relative to a bigger r module we establish an addition formula for the mean length we give two applications the first one shows that for any unital left noetherian ring r r is stably direct finite the second one shows that for any z module m the mean topological dimension of the induced action on the pontryagin dual of m coincides with the von neumann luck rank of m

homoclinic groups ie groups and expansive algebraic actions we give algebraic characterizations for expansiveness of algebraic actions of countable groups the notion of p expansiveness is introduced for algebraic actions and we show that for countable amenable groups a finitely presented algebraic action is 1 expansive exactly when it has finite entropy we also study the local entropy theory for actions of countable amenable groups on compact groups by automorphisms and show that the ie group determines the pinsker factor for such actions for an expansive algebraic action of a polycyclic by finite group on x it is shown that the entropy of the action is equal to the entropy of the induced action on the pontryagin dual of the homoclinic group the homoclinic group is a dense subgroup of the ie group the homoclinic group is nontrivial exactly when the action has positive entropy and the homoclinic group is dense in x exactly when the action has completely positive entropy

weak expansiveness for actions of sofic groups in this paper we shall introduce h expansiveness and asymptotical h expansiveness for actions of sofic groups by definition each h expansive action of a sofic group is asymptotically h expansive we show that each expansive action of a sofic group is h expansive and for any given asymptotically h expansive action of a sofic group the entropy function with respect to measures is upper semi continuous and hence the system admits a measure with maximal entropy observe that asymptotically h expansive property was first introduced and studied by misiurewicz for z actions using the language of tail entropy and thus in the remaining part of the paper we shall compare our definitions of weak expansiveness for actions of sofic groups with the definitions given in the same spirit of misiurewicz s ideas when the group is amenable it turns out that these two definitions are equivalent in this setting

a characterization of the entropies of multidimensional shifts of finite type we show that the values of entropies of multidimensional shifts of finite type sfts are characterized by a certain computation theoretic property a real number h 0 is the entropy of such an sft if and only if it is right recursively enumerable i e there is a computable sequence of rational numbers converging to h from above the same characterization holds for the entropies of sofic shifts on the other hand the entropy of strongly irreducible sfts is computable

a remark on t valued cohomology groups of algebraic group actions we prove that for a weakly mixing algebraic action g x the nth cohomology group hn g x t after quotienting out the natural subgroup hn g t contains hn g x as a natural subgroup for n 1 if we further assume the diagonal actions 2 4 are t cocycle superrigid and h2 g x is torsion free as an abelian group then the above also holds true for n 2 applying it for principal algebraic actions when n 1 we show that h2 g zg is torsion free as an abelian group when g has property t as a direct corollary of sorin popa s cocycle superrigidity theorem we also use it when n 2 to answer negatively a question of sorin popa on the 2nd cohomology group of bernoulli shift actions of property t groups

fuglede kadison determinant theme and variations abstract we review the definition of determinants for finite von neumann algebras due to fuglede and kadison fuglede b kadison r 1952 ann math 55 520 530 and a generalization for appropriate groups of invertible elements in banach algebras from a paper by skandalis and the author 1984 after some discussion of k theory and whitehead torsion we indicate the relevance of these determinants to the study of torsion in topology contents are as follows 1 the classical setting 2 on von neumann algebras and traces 3 fuglede kadison determinant for finite von neumann algebras 4 motivating question 5 brief reminder of and bott periodicity 6 revisiting the fuglede kadison and other determinants 7 on whitehead torsion 8 a few lines on torsion