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

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

on unitary representation theories of compact quantum groups categorical structure of unitary representation of compact quantum groups is studied with relation to a metrical structure encountered in the monoidal category of bimodules of finite jones index

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

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

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

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

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

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

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

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

the peculiarity of a negative coordinate axis in dyonic solutions of noncommutative n 4 super yang mills we show that in the neighborhood of a negative coordinate axis the u 1 sector of the static dyonic solutions to the noncommutative u 4 n 4 super yang mills sym can be consistently decoupled from the su 4 to all orders in the noncommutativity parameter we show the above decoupling in two ways first we show the noncommutative dyon being the same as the commutative dyon is a consistent solution to noncommutative equations of motion in the abovementioned region of noncommutative space second as an example of decoupling of a non null u 1 sector we also obtain a family of solutions with nontrivial u 1 components for all components of the gauge field in the same region of noncommutative space in both cases the su 4 and u 1 components separately satisfy the equations of motion

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

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

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