Unique Measure Of Maximal Entropy Research Articles

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.

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).

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.

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.

Measure Theoretic Entropy of Random Substitution Subshifts

Subshifts of deterministic substitutions are ubiquitous objects in dynamical systems and aperiodic order (the mathematical theory of quasicrystals). Two of their most striking features are that they have low complexity (zero topological entropy) and are uniquely ergodic. Random substitutions are a generalisation of deterministic substitutions where the substituted image of a letter is determined by a Markov process. In stark contrast to their deterministic counterparts, subshifts of random substitutions often have positive topological entropy, and support uncountably many ergodic measures. The underlying Markov process singles out one of the ergodic measures, called the frequency measure. Here, we develop new techniques for computing and studying the entropy of these frequency measures. As an application of our results, we obtain closed form formulas for the entropy of frequency measures for a wide range of random substitution subshifts and show that in many cases there exists a frequency measure of maximal entropy. Further, for a class of random substitution subshifts, we prove that this measure is the unique measure of maximal entropy. These subshifts do not satisfy Bowen’s specification property or the weaker specification property of Climenhaga and Thompson and hence provide an interesting new class of intrinsically ergodic subshifts.

Proper -colorings of are Bernoulli

AbstractWe consider the unique measure of maximal entropy for proper 3-colorings of $\mathbb {Z}^{2}$ , or equivalently, the so-called zero-slope Gibbs measure. Our main result is that this measure is Bernoulli, or equivalently, that it can be expressed as the image of a translation-equivariant function of independent and identically distributed random variables placed on $\mathbb {Z}^{2}$ . Along the way, we obtain various estimates on the mixing properties of this measure.

Equilibrium states for the random $$\beta$$- transformation through $$g$$-measures

We consider the random $$\beta$$ -transformation $$K_{\beta}$$ , defined on $$\{0,1\}^{\mathbb{N}}\times[0,\frac{\lfloor\beta\rfloor]}{\beta-1}]$$ ], that generates all possible expansions of the form $$x=\sum_{i=0}^{\infty}\frac{a_i}{\beta^i}$$ , where $$a_i\in \{0,1 , \ldots,\lfloor\beta\rfloor\}$$ }. This transformation was introduced in [3–5], where two natural invariant ergodic measures were found. The first is the unique measure of maximal entropy, and the second is a measure of the form $$m_p\times \mu_{\beta}$$ , with $$m_p$$ the Bernoulli $$(p,1-p)$$ product measure and $$\mu_{\beta}$$ is a measure equivalent to the Lebesgue measure. In this paper, we give an uncountable family of $$K_{\beta}$$ -invariant exact $$g$$ -measures for a certain collection of algebraic $$\beta$$ ’s. The construction of these $$g$$ -measures is explicit and the corresponding potentials are not locally constant.

Large deviation principle for piecewise monotonic maps with density of periodic measures

AbstractWe show that a piecewise monotonic map with positive topological entropy satisfies the level-2 large deviation principle with respect to the unique measure of maximal entropy under the conditions that the corresponding Markov diagram is irreducible and that the periodic measures of the map are dense in the set of ergodic measures. This result can apply to a broad class of piecewise monotonic maps, such as monotonic mod one transformations and piecewise monotonic maps with two monotonic pieces.

Excursions to the cusps for geometrically finite hyperbolic orbifolds and equidistribution of closed geodesics in regular covers

AbstractWe give a finitary criterion for the convergence of measures on non-elementary geometrically finite hyperbolic orbifolds to the unique measure of maximal entropy. We give an entropy criterion controlling escape of mass to the cusps of the orbifold. Using this criterion, we prove new results on the distribution of collections of closed geodesics on such an orbifold, and as a corollary, we prove the equidistribution of closed geodesics up to a certain length in amenable regular covers of geometrically finite orbifolds.

Uniqueness of the measure of maximal entropy for the standard map

In this paper we prove that for sufficiently large parameters the standard map has a unique measure of maximal entropy (m.m.e.). Moreover, we prove: the m.m.e. is Bernoulli, and the periodic points with Lyapunov exponents bounded away from zero equidistribute with respect to the m.m.e.We prove some estimates regarding the Hausdorff dimension of the m.m.e. and about the density of the support of the measure on the manifold. For a generic large parameter, we prove that the support of the m.m.e. has Hausdorff dimension 2. We also obtain the C^2 -robustness of several of these properties.

Uniqueness of the measure of maximal entropy for geodesic flows on certain manifolds without conjugate points

We prove that for closed surfaces M with Riemannian metrics without conjugate points and genus ≥2 the geodesic flow on the unit tangent bundle T1M has a unique measure of maximal entropy. Furthermore, this measure is fully supported on T1M, is the limiting distribution of closed orbits, and the flow is mixing with respect to this measure. We formulate conditions under which this result extends to higher dimensions.

On entropy and intrinsic ergodicity of coded subshifts

Any coded subshift X C X_C defined by a set C C of code words contains a subshift, which we call L C L_C , consisting of limits of single code words. We show that when C C satisfies the unique decipherability property, the topological entropy h ( X C ) h(X_C) of X C X_C is determined completely by h ( L C ) h(L_C) and the number of code words of each length. More specifically, we show that h ( X C ) = h ( L C ) h(X_C) = h(L_C) exactly when a certain infinite series is less than or equal to 1 1 , and when that series is greater than 1 1 , we give a formula for h ( X C ) h(X_C) . In the latter case, an immediate corollary (using a result from [Israel J. Math. 192 (2012), pp. 785–817] is that X C X_C has a unique measure of maximal entropy.

The unique measure of maximal entropy for a compact rank one locally CAT(0) space

<p style='text-indent:20px;'>Let <inline-formula><tex-math id="M1">\begin{document}$ X $\end{document}</tex-math></inline-formula> be a compact, geodesically complete, locally CAT(0) space such that the universal cover admits a rank one axis. We prove the Bowen-Margulis measure on the space of geodesics is the unique measure of maximal entropy for the geodesic flow, which has topological entropy equal to the critical exponent of the Poincaré series.

Measures of maximal entropy for suspension flows

We study suspension flows defined over sub-shifts of finite type with continuous roof functions. We prove the existence of suspension flows with uncountably many ergodic measures of maximal entropy. More generally, we prove that any suspension flow defined over a sub-shift of finite type can be perturbed (by an arbitrarily small perturbation) so that the resulting flow has uncountably many ergodic measures of maximal entropy, and that the same can be arranged so that the new flow has a unique measure of maximal entropy.

Uniqueness and exponential mixing for the measure of maximal entropy for piecewise hyperbolic maps

<p style='text-indent:20px;'>For a class of piecewise hyperbolic maps in two dimensions, we propose a combinatorial definition of topological entropy by counting the maximal, open, connected components of the phase space on which iterates of the map are smooth. We prove that this quantity dominates the measure theoretic entropies of all invariant probability measures of the system, and then construct an invariant measure whose entropy equals the proposed topological entropy. We prove that our measure is the unique measure of maximal entropy, that it is ergodic, gives positive measure to every open set, and has exponential decay of correlations against Hölder continuous functions. As a consequence, we also prove a lower bound on the rate of growth of periodic orbits. The main tool used in the paper is the construction of anisotropic Banach spaces of distributions on which the relevant weighted transfer operator has a spectral gap. We then construct our measure of maximal entropy by taking a product of left and right maximal eigenvectors of this operator.

Automorphisms of surfaces: Kummer rigidity and measure of maximal entropy

We classify complex projective surfaces X with an automorphism f of positive entropy for which the unique measure of maximal entropy is absolutely continuous with respect to the Lebesgue measure. As a byproduct, if X is a K3 surface and is not a Kummer surface, the periodic points of f are equidistributed with respect to a probability measure which is singular with respect to the canonical volume of X . The proof is based on complex algebraic geometry and Hodge theory, Pesin’s theory and renormalization techniques. A crucial argument relies on a new compactness property of entire curves parametrizing the invariant manifolds of the automorphism.

On the measure of maximal entropy for finite horizon Sinai Billiard maps

The Sinai billiard map T T on the two-torus, i.e., the periodic Lorentz gas, is a discontinuous map. Assuming finite horizon, we propose a definition h ∗ h_* for the topological entropy of T T . We prove that h ∗ h_* is not smaller than the value given by the variational principle, and that it is equal to the definitions of Bowen using spanning or separating sets. Under a mild condition of sparse recurrence to the singularities, we get more: First, using a transfer operator acting on a space of anisotropic distributions, we construct an invariant probability measure μ ∗ \mu _* of maximal entropy for T T (i.e., h μ ∗ ( T ) = h ∗ h_{\mu _*}(T)=h_* ), we show that μ ∗ \mu _* has full support and is Bernoulli, and we prove that μ ∗ \mu _* is the unique measure of maximal entropy and that it is different from the smooth invariant measure except if all nongrazing periodic orbits have multiplier equal to h ∗ h_* . Second, h ∗ h_* is equal to the Bowen–Pesin–Pitskel topological entropy of the restriction of T T to a noncompact domain of continuity. Last, applying results of Lima and Matheus, as upgraded by Buzzi, the map T T has at least C e n h ∗ C e^{nh_*} periodic points of period n n for all n ∈ N n \in \mathbb {N} .

Fluctuations of ergodic sums on periodic orbits under specification

<p style='text-indent:20px;'>We study the fluctuations of ergodic sums using global and local specifications on periodic points. We obtain Lindeberg-type central limit theorems in both situations. As an application, when the system possesses a unique measure of maximal entropy, we show weak convergence of ergodic sums to a mixture of normal distributions. Our results suggest decomposing the variances of ergodic sums according to global and local sources.