Measure Of Maximal Entropy Research Articles

On a Degenerating Limit Theorem of DeMarco–Faber

One of our aims is to complement the proof of DeMarco–Faber’s degenerating limit theorem for the family of the unique maximal entropy measures parametrized by a punctured open disk associated to a meromorphic family of rational functions on the complex projective line, degenerating at the puncture. This complementation is done by our main result, which rectifies a key computation in their argument. We also establish and use a direct and explicit translation from degenerating complex dynamics into quantized Berkovich dynamics, instead of using DeMarco–Faber’s more conceptual transfer principle between those dynamics.

Fourier decay of equilibrium states for bunched attractors

Let M be a closed manifold, and let f:M→M be a C2+α Axiom A diffeomorphism. Suppose that f has an attractor Ω with codimension 1 stable lamination. Under a generic nonlinearity condition and a suitable bunching condition, we prove polynomial Fourier decay in the unstable direction for a large class of invariant measures on Ω. Our result applies in particular for the measure of maximal entropy. We construct in the appendix an explicit solenoid that satisfies the nonlinearity and bunching assumption.

Equilibrium States of Endomorphisms of \\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$\\mathbb{P}^{k}$\\end{document}: Spectral Stability and Limit Theorems

We establish the existence of a spectral gap for the transfer operator induced on \\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$\\mathbb{P}^{k} = \\mathbb{P}^{k} (\\mathbb{C})$\\end{document} by a generic holomorphic endomorphism and a suitable continuous weight and its perturbations on various functional spaces, which is new even in dimension one. Thanks to the spectral gap, we establish an exponential speed of convergence for the equidistribution of the backward orbits of points towards the conformal measure and the exponential mixing. Moreover, as an immediate consequence, we obtain a full list of statistical properties for the equilibrium states: CLT, Berry-Esseen Theorem, local CLT, ASIP, LIL, LDP, almost sure CLT. Many of these properties are new even in dimension one, some even in the case of zero weight function (i.e., for the measure of maximal entropy).

Similarities and differences between specification and non-uniform specification

Abstract Pavlov [Adv. Math.295 (2016), 250–270; Nonlinearity32 (2019), 2441–2466] studied the measures of maximal entropy for dynamical systems with weak versions of specification property and found the existence of intrinsic ergodicity would be influenced by the assumptions of the gap functions. Inspired by these, in this article, we study the dynamical systems with non-uniform specification property. We give some basic properties these systems have and give an assumption for the gap functions to ensure the systems have the following five properties: CO-measures are dense in invariant measures; for every non-empty compact connected subset of invariant measures, its saturated set is dense in the total space; ergodic measures are residual in invariant measures; ergodic measures are connected; and entropy-dense. In addition, we will give examples to show the assumption is optimal.

Geodesic flows of compact higher genus surfaces without conjugate points have expansive factors

In this paper we show that a geodesic flow of a compact surface without conjugate points of genus greater than one is time-preserving semi-conjugate to a continuous expansive flow which is topologically mixing and has a local product structure. As an application we show that the geodesic flow of a compact surface without conjugate points of genus greater than one has a unique measure of maximal entropy. This gives an alternative proof of Climenhaga–Knieper–War Theorem.

Shifts of finite type on locally finite groups

Abstract In this work we prove that every shift of finite type (SFT), sofic shift, and strongly irreducible shift on locally finite groups has strong dynamical properties. These properties include that every sofic shift is an SFT, every SFT is strongly irreducible, every strongly irreducible shift is an SFT, every SFT is entropy minimal, and every SFT has a unique measure of maximal entropy, among others. In addition, we show that if every SFT on a group is strongly irreducible, or if every sofic shift is an SFT, then the group must be locally finite, and this extends to all of the properties we explore. These results are collected in two main theorems which characterize the local finiteness of groups by purely dynamical properties. In pursuit of these results, we present a formal construction of free extension shifts on a group G, which takes a shift on a subgroup H of G, and naturally extends it to a shift on all of G.

Measures of maximal entropy of bounded density shifts

Abstract We find sufficient conditions for bounded density shifts to have a unique measure of maximal entropy. We also prove that every measure of maximal entropy of a bounded density shift is fully supported. As a consequence of this, we obtain that bounded density shifts are surjunctive.

Every complex Hénon map is exponentially mixing of all orders and satisfies the CLT

Abstract We show that the measure of maximal entropy of every complex Hénon map is exponentially mixing of all orders for Hölder observables. As a consequence, the Central Limit Theorem holds for all Hölder observables.

Density of periodic measures and large deviation principle for generalised mod one transformations

We introduce a new class of piecewise monotonic maps, called generalised mod one transformations, which include all (α,β) and generalised β -transformations, and prove that all transitive generalised mod one transformations with positive topological entropy satisfy the level-2 large deviation principle with a unique measure of maximal entropy. This is obtained by our result on the density of periodic measures for more general class of piecewise mototonic maps, which is a partial answer to the open problem posed by Hofbauer and Raith (1998 Fund. Math. 157 221–34); Raith (www.math.iupui.edu/~mmisiure/open/PR1.pdf).

Rates of mixing for the measure of maximal entropy of dispersing billiard maps

AbstractIn a recent work, Baladi and Demers constructed a measure of maximal entropy for finite horizon dispersing billiard maps and proved that it is unique, mixing and moreover Bernoulli. We show that this measure enjoys natural probabilistic properties for Hölder continuous observables, such as at least polynomial decay of correlations and the Central Limit Theorem. The results of Baladi and Demers are subject to a condition of sparse recurrence to singularities. We use a similar and slightly stronger condition, and it has a direct effect on our rate of decay of correlations. For billiard tables with bounded complexity (a property conjectured to be generic), we show that the sparse recurrence condition is always satisfied and the correlations decay at a super‐polynomial rate.

On amenability and measure of maximal entropy for semigroups of rational maps: II

We compare dynamical and algebraic properties of semigroups of rational maps. In particular, we show a version of the Day-von Neumann conjecture and give a partial positive answer to “Sushkievich’s problem” for semigroups of rational maps. We relate these conjectures with Furstenberg’s [Formula: see text] problem and prove a coarse version of Furstenberg’s problem for semigroups of non-exceptional polynomials.

Hyperbolicity of maximal entropy measures for certain maps isotopic to Anosov diffeomorphisms

We prove the hyperbolicity of ergodic maximal entropy measures for a class of partially hyperbolic diffeomorphisms of Td, which have a compact two-dimensional center foliation.

Counting independent sets in amenable groups

AbstractGiven a locally finite graph $\Gamma $ , an amenable subgroup G of graph automorphisms acting freely and almost transitively on its vertices, and a G-invariant activity function $\unicode{x3bb} $ , consider the free energy $f_G(\Gamma ,\unicode{x3bb} )$ of the hardcore model defined on the set of independent sets in $\Gamma $ weighted by $\unicode{x3bb} $ . Under the assumption that G is finitely generated and its word problem can be solved in exponential time, we define suitable ensembles of hardcore models and prove the following: if $\|\unicode{x3bb} \|_\infty < \unicode{x3bb} _c(\Delta )$ , there exists a randomized $\epsilon $ -additive approximation scheme for $f_G(\Gamma ,\unicode{x3bb} )$ that runs in time $\mathrm {poly}((1+\epsilon ^{-1})\lvert \Gamma /G \rvert )$ , where $\unicode{x3bb} _c(\Delta )$ denotes the critical activity on the $\Delta $ -regular tree. In addition, if G has a finite index linearly ordered subgroup such that its algebraic past can be decided in exponential time, we show that the algorithm can be chosen to be deterministic. However, we observe that if $\|\unicode{x3bb} \|_\infty> \unicode{x3bb} _c(\Delta )$ , there is no efficient approximation scheme, unless $\mathrm {NP} = \mathrm {RP}$ . This recovers the computational phase transition for the partition function of the hardcore model on finite graphs and provides an extension to the infinite setting. As an application in symbolic dynamics, we use these results to develop efficient approximation algorithms for the topological entropy of subshifts of finite type with enough safe symbols, we obtain a representation formula of pressure in terms of random trees of self-avoiding walks, and we provide new conditions for the uniqueness of the measure of maximal entropy based on the connective constant of a particular associated graph.

A relation between entropy and transitivity of Anosov diffeomorphisms

It is known that transitive Anosov diffeomorphisms have a unique measure of maximal entropy (MME). Here we discuss the converse question. Transitivity of Anosov diffeomorphisms can be reached under suitable hypotheses on Lyapunov exponents on the set of periodic points and the structure of the MME. In another way, assuming together the uniqueness of MME and that every point is regular, in Oseledec’s Theorem sense, also we can get the transitivity of Anosov diffeomorphisms in this setting.

On the Correspondence between Subshifts of Finite Type and Statistical Mechanics Models.

Several classical problems in symbolic dynamics concern the characterization of the simplex of measures of maximal entropy. For subshifts of finite type in higher dimensions, methods of statistical mechanics are ideal for dealing with these problems. R. Burton and J. Steif developed a strategy to construct examples of strongly irreducible subshifts of finite type admitting several measures of maximal entropy. This strategy exploits a correspondence between equilibrium statistical mechanics and symbolic dynamics-a correspondence which was later formalized by O. Häggström. In this paper, we revisit and discuss this correspondence with the aim of presenting a simplified version of it and present some applications of rigorous results concerning the Potts model and the six-vertex model to symbolic dynamics, illustrating in this way the possibilities of this correspondence.

Effective intrinsic ergodicity for countable state Markov shifts

For strongly positively recurrent countable state Markov shifts, we bound the distance between an invariant measure and the measure of maximal entropy in terms of the difference of their entropies. This extends an earlier result for subshifts of finite type, due to Kadyrov. We provide a similar bound for equilibrium measures of strongly positively recurrent potentials, in terms of the pressure difference. For measures with nearly maximal entropy, we have new, and sharp, bounds. The strong positive recurrence condition is necessary.

The Lanford–Ruelle theorem for actions of sofic groups

Let Γ \Gamma be a sofic group, Σ \Sigma be a sofic approximation sequence of Γ \Gamma and X X be a Γ \Gamma -subshift with non-negative sofic topological entropy with respect to Σ \Sigma . Further assume that X X is a shift of finite type, or more generally, that X X satisfies the topological Markov property. We show that for any sufficiently regular potential f : X → R f \colon X \to \mathbb {R} , any translation-invariant Borel probability measure on X X which maximizes the measure-theoretic sofic pressure of f f with respect to Σ \Sigma is a Gibbs state with respect to f f . This extends a classical theorem of Lanford and Ruelle, as well as previous generalizations of Moulin Ollagnier, Pinchon, Tempelman and others, to the case where the group is sofic. As applications of our main result we present a criterion for uniqueness of an equilibrium measure, as well as sufficient conditions for having that the equilibrium states do not depend upon the chosen sofic approximation sequence. We also prove that for any group-shift over a sofic group, the Haar measure is the unique measure of maximal sofic entropy for every sofic approximation sequence, as long as the homoclinic group is dense. On the expository side, we present a short proof of Chung’s variational principle for sofic topological pressure.

Statistical properties of periodic points for infinitely renormalizable unimodal maps

For an infinitely renormalizable negative Schwarzian unimodal map f with a non-flat critical point, we analyze statistical properties of periodic points as their periods tend to infinity. Since the standard sequence of probability measures constructed from periodic points weighted with Birkhoff sums of a given potential does not always converge to an equilibrium state, we consider another sequence of probability measures obtained by averaging over certain time windows. For a weight φ which is a continuous function or a geometric potential −β log|f′|, we obtain level-2 large deviation bounds. From the upper bound, we deduce that weighted periodic points asymptotically distribute with respect to equilibrium states for the potential φ. It follows that periodic points asymptotically distribute with respect to measures of maximal entropy, and periodic points weighted with their Lyapunov exponents asymptotically distribute with respect to the post-critical measure supported on the attracting Cantor set. In the case the pressure of φ is non-positive, we obtain the level-2 large deviation principle.

Existence and uniqueness of equilibrium states for systems with specification at a fixed scale: an improved Climenhaga–Thompson criterion* *This research has been supported [in part] by CAPES—Finance Code 001 and CNPq-grants. MJP was partially supported by Grant Cientista do Nosso Estado-FAPERJ.

We consider the uniqueness of equilibrium states for dynamical systems that satisfy certain weak, non-uniform versions of specification, expansivity, and the Bowen property at a fixed scale. Following Climenhaga–Thompson’s approach which was originally due to Bowen and Franco, we prove that equilibrium states are unique even when the weak specification assumption only holds on a small collection of orbit segments. This improvement will be crucial in a subsequent work, where we will prove that (open and densely) every Lorenz attractor supports a unique measure of maximal entropy.

Julia Sets of Hyperbolic Rational Maps have Positive Fourier Dimension

Let \(f:\widehat{{\mathbb {C}}}\rightarrow \widehat{{\mathbb {C}}}\) be a hyperbolic rational map of degree \(d \ge 2\), and let \(J \subset {\mathbb {C}}\) be its Julia set. We prove that J always has positive Fourier dimension. The case where J is included in a circle follows from a recent work of Sahlsten and Stevens (Fourier transform and expanding maps on Cantor sets preprint. arXiv:2009.01703, 2020). In the case where J is not included in a circle, we prove that a large family of probability measures supported on J exhibit polynomial Fourier decay: our result applies in particular to the measure of maximal entropy and to the conformal measure.