Tom Meyerovitch

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

FINITE ENTROPY FOR MULTIDIMENSIONAL CELLULAR AUTOMATA

hgeq 0

Poisson suspensions and entropy for infinite transformations

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

Harmonic functions of linear growth on solvable groups

h

Gibbs and equilibrium measures for some families of subshifts

from above. The same characterization holds for the entropies of sofic shifts. On the other hand, the entropy of an irreducible SFT is computable.

Markov random fields, Markov cocycles and the 3-colored chessboard

Let X = SG where G is a countable group and S is a finite set. A cellular automaton (CA) is an endomorphism T : X ! X (continuous, com- muting with the action of G). Shereshevsky (14) proved that for G = Z d with d > 1 no CA can be forward expansive, raising the following conjecture: For G = Z d , d > 1 the topological entropy of any CA is either zero or infinite. Mor- ris and Ward (11), proved this for linear CAs, leaving the original conjecture open. We show that this conjecture is false, proving that for any d there exist a d-dimensional CA with finite, nonzero topological entropy. We also discuss a measure-theoretic counterpart of this question for measure-preserving CAs.

Encoding semiconstrained systems

The Poisson entropy of an infinite-measure-preserving transformation is defined as the Kolmogorov entropy of its Poisson suspension. In this article, we relate Poisson entropy with other definitions of entropy for infinite transformations: For quasi-finite transformations we prove that Poisson entropy coincides with Krengels and Parrys entropy. In particular, this implies that for null-recurrent Markov chains, the usual formula for the entropy

Pseudo-orbit tracing and algebraic actions of countable amenable groups

-sum q_i p_{i,j}log p_{i,j}

Multidimensional semiconstrained systems

holds in any of the definitions for entropy. Poisson entropy dominates Parrys entropy in any conservative transformation. We also prove that relative entropy (in the sense of Danilenko and Rudolph) coincides with the relative Poisson entropy. Thus, for any factor of a conservative transformation, difference of the Krengels entropy is equal to the difference of the Poisson entropies. In case there exists a factor with zero Poisson entropy, we prove the existence of a maximum (Pinsker) factor with zero Poisson entropy. Together with the preceding results, this answers affirmatively the question raised in arXiv:0705.2148v3 about existence of a Pinsker factor in the sense of Krengel for quasi-finite transformations.

Direct topological factorization for topological flows

In this work we study the structure of finitely generated groups for which a space of harmonic functions with fixed polynomial growth is finite dimensional. It is conjectured that such groups must be virtually nilpotent (the converse direction to Kleiner’s theorem). We prove that this is indeed the case for solvable groups. The investigation is partly motivated by Kleiner’s proof for Gromov’s theorem on groups of polynomial growth.