arXiv subjects

Tom Meyerovitch

Publications and source records attributed to Tom Meyerovitch.

At least 37 records · Page 2Linked to original sources

arXiv2020

Borel subsystems and ergodic universality for compact $\mathbb Z^d$-systems via specification and beyond

A Borel system $(X,S)$ is `almost Borel universal' if any free Borel dynamical system $(Y,T)$ of strictly lower entropy is isomorphic to a Borel subsystem of $(X,S)$, after removing a null set. We obtain and exploit a new sufficient condition for a topological dynamical system to be almost Borel universal. We use our main result to deduce various conclusions and answer a number of questions. Along with additional results, we prove that a `generic' homeomorphism of a compact manifold of topological dimension at least two can model any ergodic transformation, that non-uniform specification implies almost Borel universality, and that $3$-colorings in $\mathbb Z^d$ and dimers in $\mathbb Z^2$ are almost Borel universal

math.DS↗

arXiv2020

Gibbsian representations of continuous specifications: the theorems of Kozlov and Sullivan revisited

The theorems of Kozlov and Sullivan characterize Gibbs measures as measures with positive continuous specifications. More precisely, Kozlov showed that every positive continuous specification on symbolic configurations of the lattice is generated by a norm-summable interaction. Sullivan showed that every shift-invariant positive continuous specification is generated by a shift-invariant interaction satisfying the weaker condition of variation-summability. These results were proven in the 1970s. An open question since that time is whether Kozlov's theorem holds in the shift-invariant setting, equivalently whether Sullivan's conclusion can be improved from variation-summability to norm-summability. We show that the answer is no: there exist shift-invariant positive continuous specifications that are not generated by any shift-invariant norm-summable interaction. On the other hand, we give a complete proof of an extension, suggested by Kozlov, of Kozlov's theorem to a characterization of positive continuous specifications on configuration spaces with arbitrary hard constraints. We also present an extended version of Sullivan's theorem. Aside from simplifying some of the arguments in the original proof, our new version of Sullivan's theorem applies in various settings not covered by the original proof. In particular, it applies when the support of the specification is the hard-core shift or the two-dimensional $q$-coloring shift for $q\geq 6$.

math-ph↗

arXiv2019

Predictability, topological entropy and invariant random orders

We prove that a topologically predictable action of a countable amenable group has zero topological entropy, as conjectured by Hochman. On route, we investigate invariant random orders and formulate a unified Kieffer-Pinsker formula for the Kolmogorov-Sinai entropy of measure preserving actions of amenable groups. We also present a proof due to Weiss for the fact that topologically prime actions of sofic groups have non-positive topological sofic entropy.

math.DS↗

arXiv2017

Expansive multiparameter actions and mean dimension

Mañé (1979) proved that if a compact metric space admits an expansive homeomorphism then it is finite dimensional. We generalize this theorem to multiparameter actions. The generalization involves mean dimension theory, which counts "averaged dimension" of a dynamical system. We prove that if $T:\mathbb{Z}^k\times X\to X$ is expansive and if $R:\mathbb{Z}^{k-1}\times X\to X$ commutes with $T$ then $R$ has finite mean dimension. When $k=1$, this statement reduces to Mañé's theorem. We also study several related issues, especially the connection with entropy theory.

math.DS↗

arXiv2017

On Independence and Capacity of Multidimensional Semiconstrained Systems

We find a new formula for the limit of the capacity of certain sequences of multidimensional semiconstrained systems as the dimension tends to infinity. We do so by generalizing the notion of independence entropy, originally studied in the context of constrained systems, to the study of semiconstrained systems. Using the independence entropy, we obtain new lower bounds on the capacity of multidimensional semiconstrained systems in general, and $d$-dimensional axial-product systems in particular. In the case of the latter, we prove our bound is asymptotically tight, giving the exact limiting capacity in terms of the independence entropy. We show the new bound improves upon the best-known bound in a case study of $(0,k,p)$-RLL.

math.DS↗

arXiv2017

On pointwise periodicity in tilings, cellular automata and subshifts

We study implications of expansiveness and pointwise periodicity for certain groups and semigroups of transformations. Among other things we prove that every pointwise periodic finitely generated group of cellular automata is necessarily finite. We also prove that a subshift over any finitely generated group that consists of finite orbits is finite, and related results for tilings of Euclidean space.

math.DS↗

arXiv2017

Pseudo-Orbit Tracing and Algebraic actions of countable amenable groups

Consider a countable amenable group acting by homeomorphisms on a compact metrizable space. Chung and Li asked if expansiveness and positive entropy of the action imply existence of an off-diagonal asymptotic pair. For algebraic actions of polycyclic-by-finite groups, Chung and Li proved it does. We provide examples showing that Chung and Li's result is near-optimal in the sense that the conclusion fails for some non-algebraic action generated by a single homeomorphism, and for some algebraic actions of non-finitely generated abelian groups. On the other hand, we prove that every expansive action of an amenable group with positive entropy that has the pseudo-orbit tracing property must admit off-diagonal asymptotic pairs. Using Chung and Li's algebraic characterization of expansiveness, we prove the pseudo-orbit tracing property for a class of expansive algebraic actions. This class includes every expansive principal algebraic action of an arbitrary countable group.

math.DS↗

arXiv2016

Polynomials and harmonic functions on discrete groups

Alexopoulos proved that on a finitely generated virtually nilpotent group, the restriction of a harmonic function of polynomial growth to a torsion-free nilpotent subgroup of finite index is always a polynomial in the Mal'cev coordinates of that subgroup. For general groups, vanishing of higher-order discrete derivatives gives a natural notion of polynomial maps, which has been considered by Leibman and others. We provide a simple proof of Alexopoulos's result using this notion of polynomials, under the weaker hypothesis that the space of harmonic functions of polynomial growth of degree at most k is finite dimensional. We also prove that for a finitely generated group the Laplacian maps the polynomials of degree k surjectively onto the polynomials of degree k - 2. We then present some corollaries. In particular, we calculate precisely the dimension of the space of harmonic functions of polynomial growth of degree at most k on a virtually nilpotent group, extending an old result of Heilbronn for the abelian case, and refining a more recent result of Hua and Jost.

math.GR↗

arXiv2016

Encoding Semiconstrained Systems

Semiconstrained systems were recently suggested as a generalization of constrained systems, commonly used in communication and data-storage applications that require certain offending subsequences be avoided. In an attempt to apply techniques from constrained systems, we study sequences of constrained systems that are contained in, or contain, a given semiconstrained system, while approaching its capacity. In the case of contained systems we describe to such sequences resulting in constant-to-constant bit-rate block encoders and sliding-block encoders. Surprisingly, in the case of containing systems we show that a "generic" semiconstrained system is never contained in a proper fully-constrained system.

cs.IT↗

arXiv2016

Positive sofic entropy implies finite stabilizer

We prove that for a measure preserving action of a sofic group with positive sofic entropy, the set of points with finite stabilizer have positive measure. This extends results of Weiss and Seward for amenable groups and free groups, respectively. It follows that the action of a sofic group on its subgroups by inner automorphisms has zero topological sofic entropy, and a faithful action with completely positive sofic entropy must be free.

math.DS↗

arXiv2015

Harmonic functions of linear growth on solvable groups

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.

math.GR↗

arXiv2015

Markov Random Fields, Markov Cocycles and The 3-colored Chessboard

The well-known Hammersley-Clifford theorem states (under certain conditions) that any Markov random field is a Gibbs state for a nearest neighbor interaction. In this paper we study Markov random fields for which the proof of the Hammersley-Clifford theorem does not apply. Following Petersen and Schmidt we utilize the formalism of cocycles for the homoclinic equivalence relation and introduce "Markov cocycles", reparametrisations of Markov specifications. The main part of this paper exploits this to deduce the conclusion of the Hammersley-Clifford theorem for a family of Markov fields which are outside the theorem's purview where the underlying graph is $\mathbb{Z}^d$. This family includes all Markov random fields whose support is the d-dimensional "3-colored chessboard". On the other extreme, we construct a family of shift-invariant Markov random fields which are not given by any finite range shift-invariant interaction.

math.DS↗

arXiv2015

Direct topological factorization for topological flows

This paper considers the general question of when a topological action of a countable group can be factored into a direct product of a nontrivial actions. In the early 1980's D. Lind considered such questions for $\mathbb{Z}$-shifts of finite type. We study in particular direct factorizations of subshifts of finite type over $\mathbb{Z}^d$ and other groups, and $\mathbb{Z}$-subshifts which are not of finite type. The main results concern direct factors of the multidimensional full $n$-shift, the multidimensional $3$-colored chessboard and the Dyck shift over a prime alphabet. A direct factorization of an expansive $\mathbb{G}$-action must be finite, but a example is provided of a non-expansive $\mathbb{Z}$-action for which there is no finite direct prime factorization. The question about existence of direct prime factorization of expansive actions remains open, even for $\mathbb{G}=\mathbb{Z}$.

math.DS↗

arXiv2015

Semi-constrained Systems

When transmitting information over a noisy channel, two approaches, dating back to Shannon's work, are common: assuming the channel errors are independent of the transmitted content and devising an error-correcting code, or assuming the errors are data dependent and devising a constrained-coding scheme that eliminates all offending data patterns. In this paper we analyze a middle road, which we call a semiconstrained system. In such a system, which is an extension of the channel with cost constraints model, we do not eliminate the error-causing sequences entirely, but rather restrict the frequency in which they appear. We address several key issues in this study. The first is proving closed-form bounds on the capacity which allow us to bound the asymptotics of the capacity. In particular, we bound the rate at which the capacity of the semiconstrained $(0,k)$-RLL tends to $1$ as $k$ grows. The second key issue is devising efficient encoding and decoding procedures that asymptotically achieve capacity with vanishing error. Finally, we consider delicate issues involving the continuity of the capacity and a relaxation of the definition of semiconstrained systems.

cs.IT↗

arXiv2013

Ergodicity of Poisson products and applications

In this paper we study the Poisson process over a $σ$-finite measure-space equipped with a measure preserving transformation or a group of measure preserving transformations. For a measure-preserving transformation $T$ acting on a $σ$-finite measure-space $X$, the Poisson suspension of $T$ is the associated probability preserving transformation $T_*$ which acts on realization of the Poisson process over $X$. We prove ergodicity of the Poisson-product $T\times T_*$ under the assumption that $T$ is ergodic and conservative. We then show, assuming ergodicity of $T\times T_*$, that it is impossible to deterministically perform natural equivariant operations: thinning, allocation or matching. In contrast, there are well-known results in the literature demonstrating the existence of isometry equivariant thinning, matching and allocation of homogenous Poisson processes on $\mathbb{R}^d$. We also prove ergodicity of the "first return of left-most transformation" associated with a measure preserving transformation on $\mathbb{R}_+$, and discuss ergodicity of the Poisson-product of measure preserving group actions, and related spectral properties.

math.DS↗

arXiv2013

On independence and entropy for high-dimensional isotropic subshifts

In this work, we study the problem of finding the asymptotic growth rate of the number of of $d$-dimensional arrays with side length $n$ over a given alphabet which avoid a list of one-dimensional "forbidden" words along all cardinal directions, as both $n$ and $d$ tend to infinity. Louidor, Marcus, and the second author called this quantity the "limiting entropy"; it is the limit of a sequence of topological entropies of a sequence of isotropic $\mathbb{Z}^d$ subshifts with the dimension $d$ tending to infinity. We find an expression for this limiting entropy which involves only one-dimensional words, which was implicitly conjectured earlier, and given the name "independence entropy." In the case where the list of "forbidden" words is finite, this expression is algorithmically computable and is of the form $\frac{1}{n} \log k$ for $k,n \in \mathbb{N}$. Our proof also characterizes the weak limits (as $d \rightarrow \infty$) of isotropic measures of maximal entropy; any such measure is a Bernoulli extension over some zero entropy factor from an explicitly defined set of measures. We also demonstrate how our results apply to various models previously studied in the literature, in some cases recovering or generalizing known results, but in other cases proving new ones.

math.DS↗

arXiv2011

One dimensional Markov random fields, Markov chains and Topological Markov fields

In this paper we show that any one-dimensional stationary, finite-valued Markov Random Field (MRF) is a Markov chain, without any mixing condition or condition on the support. Our proof makes use of two properties of the support $X$ of a finite-valued stationary MRF: 1) $X$ is non-wandering (this is a property of the support of any finite-valued stationary process) and 2) $X$ is a topological Markov field (TMF). The latter is a new property that sits in between the classes of shifts of finite type and sofic shifts, which are well-known objects of study in symbolic dynamics. Here, we develop the TMF property in one dimension, and we will develop this property in higher dimensions in a future paper. While we are mainly interested in discrete-time finite-valued stationary MRF's, we also consider continuous-time, finite-valued stationary MRF's, and show that these are (continuous-time) Markov chains as well.

math.DS↗

arXiv2009

Stationary map coloring

We consider a planar Poisson process and its associated Voronoi map. We show that there is a proper coloring with 6 colors of the map which is a deterministic isometry-equivariant function of the Poisson process. As part of the proof we show that the 6-core of the corresponding Delaunay triangulation is empty. Generalizations, extensions and some open questions are discussed.

math.PR↗