Measure-theoretic Entropy Research Articles

Directional Pinsker algebra and its applications

In this paper, we introduce the directional Pinsker algebra, and construct a skew product to study it. As applications, we show that (i) if a Z2-system with positive directional measure-theoretic entropy then it is multivariant directional mean Li–Yorke chaotic along the corresponding direction; (ii) for any ergodic measure on a Z2-system, the intersection of the set of directional measure-theoretic entropy tuples with the set of directional asymptotic tuples is dense in the set of directional measure-theoretic entropy tuples.

A Mañé-Manning formula for expanding measures for endomorphisms of ℙ^{𝕜}

Let k ≥ 1 k \ge 1 be an integer and f f a holomorphic endomorphism of P k ( C ) \mathbb {P}^k (\mathbb {C}) of algebraic degree d ≥ 2 d\geq 2 . We introduce a dynamical volume dimension for ergodic f f -invariant probability measures with strictly positive Lyapunov exponents. In particular, this class of measures includes all ergodic measures whose measure-theoretic entropy is strictly larger than ( k − 1 ) log ⁡ d (k-1)\log d , a natural generalization of the class of measures of positive measure-theoretic entropy in dimension 1. The volume dimension is equivalent to the Hausdorff dimension when k = 1 k=1 , but depends on the dynamics of f f to incorporate the absence of an analogue of Koebe’s theorem and the non-conformality of holomorphic endomorphisms for k ≥ 2 k\geq 2 . If ν \nu is an ergodic f f -invariant probability measure with strictly positive Lyapunov exponents, we prove a generalization of the Mañé-Manning formula relating the volume dimension, the measure-theoretic entropy, and the sum of the Lyapunov exponents of ν \nu . As a consequence, we give a characterization of the first zero of a natural pressure function for such expanding measures in terms of their volume dimensions. For hyperbolic maps, such zero also coincides with the volume dimension of the Julia set, and with the exponent of a natural (volume-)conformal measure. This generalizes results by Denker-Urbański [Nonlinearity 4 (1991), pp. 365–384; Trans. Amer. Math. Soc. 328 (1991), pp. 563–587] and McMullen [Comment. Math. Helv. 75 (2000), pp. 535–593] in dimension 1 to any dimension k ≥ 1 k\geq 1 . Our methods mainly rely on a theorem by Berteloot-Dupont-Molino [Ann. Inst. Fourier (Grenoble) 58 (2008), pp. 2137–2168], which gives a precise control on the distortion of inverse branches of endomorphisms along generic inverse orbits with respect to measures with strictly positive Lyapunov exponents.

Strong Probabilistic Stability in Holomorphic Families of Endomorphisms of ℙk (ℂ) and Polynomial-Like Maps

Abstract We prove that, in stable families of endomorphisms of $\mathbb{P}^{k} (\mathbb C)$, the measurable holomorphic motion of the Julia sets introduced by Berteloot, Dupont, and the first author is unbranched at almost every point with respect to all measures on the Julia set with strictly positive Lyapunov exponents and not charging the post-critical set. This provides a parallel in this setting to the probabilistic stability of Hénon maps by Berger–Dujardin–Lyubich. An analogous result holds in families of polynomial-like maps of large topological degree. In this case, we also give a sufficient condition for the positivity of the Lyapunov exponents of an ergodic measure in terms of its measure-theoretic entropy, generalizing to this setting an analogous result by de Thélin and Dupont valid on $\mathbb{P}^{k} (\mathbb C)$.

Multifractal Analysis of Measures Arising from Random Substitutions

We study regularity properties of frequency measures arising from random substitutions, which are a generalisation of (deterministic) substitutions where the substituted image of each letter is chosen independently from a fixed finite set. In particular, for a natural class of such measures, we derive a closed-form analytic formula for the Lq\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$L^q$$\\end{document}-spectrum and prove that the multifractal formalism holds. This provides an interesting new class of measures satisfying the multifractal formalism. More generally, we establish results concerning the Lq\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$L^q$$\\end{document}-spectrum of a broad class of frequency measures. We introduce a new notion called the inflation wordLq\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$L^q$$\\end{document}-spectrum of a random substitution and show that this coincides with the Lq\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$L^q$$\\end{document}-spectrum of the corresponding frequency measure for all q≥0\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$q \\ge 0$$\\end{document}. As an application, we obtain closed-form formulas under separation conditions and recover known results for topological and measure theoretic entropy.

Sequence entropy and mean sequence dimension for non-compact metric spaces

In this paper, we introduce an extension of topological sequence entropy and d-sequence entropy for a dynamical system on a non-compact metric space. Then we give some elementary properties of those sequence entropy. In particular, we establish a variational principle which states that for a class of sequences, the supremum of the measure-theoretic sequence entropies and the minimum of the d-sequence entropies always coincide, and they are not less than the topological sequence entropy. Finally, we introduce the concepts of mean sequence dimension and maximal pattern mean topological dimension and give some properties of them.

Rotational entropy − a homotopy invariant for torus maps

To connect topological entropy with rotation theory, Botelho [2] introduced the notion of topological rotational entropy for annulus maps which are homotopic to the identity and later Geller and Misiurewicz [5] showed that it indeed vanished. In this paper, we generalize the notion of rotational entropy to any torus map, give its calculation formula and hence show that it is a homotopy invariant. We also introduce a notion of measure-theoretic rotational entropy and show that it is always less than or equal to the rotational entropy.

Measure-theoretic sequence entropy pairs and mean sensitivity

AbstractWe characterize measure-theoretic sequence entropy pairs of continuous actions of abelian groups using mean sensitivity. This addresses an open question of Li and Yu [On mean sensitive tuples. J. Differential Equations297 (2021), 175–200]. As a consequence of our results, we provide a simpler characterization of Kerr and Li’s independence sequence entropy pairs ( $\mu $ -IN-pairs) when the measure is ergodic and the group is abelian.

Topological pressure for conservative C 1-diffeomorphisms with no dominated splitting

We prove three formulas for computing the topological pressure of C 1 -generic conservative diffeomorphism with no dominated splitting and show the continuity of topological pressure with respect to these diffeomorphisms. We prove for these generic diffeomorphisms that there are no equilibrium states with positive measure theoretic entropy. In particular, for hyperbolic potentials, there are no equilibrium states. For C 1 generic conservative diffeomorphism on compact surfaces with no dominated splitting and ϕ m ( x ) := − 1 m log ⁡ ‖ D x f m ‖ , m ∈ N , we show that there exist equilibrium states with zero entropy and there exists a transition point t 0 for the one parameter family { t ϕ m } t ≥ 0 , such that there is no equilibrium states for t ∈ [ 0 , t 0 ) and there is an equilibrium state for t ∈ [ t 0 , + ∞ ) .

Topological chaos and statistical triviality

While transferring one flow ϕ with fixed points homeomorphically to its equivalent flow ψ, we show that positive topological entropy degenerates to zero (such degeneracy happens for equivalent continuous flows [3] and equivalent differential flows [6]) if and only if all ergodic invariant measures with positive measure-theoretic entropy degenerate to fixed points. Whenever ϕ is assumed to be topological transitive, the measure degeneracy implies that the resulted equivalent flow ψ is topologically chaotic but statistically trivial, meaning that all ergodic invariant measures are supported on fixed points. Using different approaches in different areas people constructed examples of topological chaotic but statistical trivial systems, see [3] for C0 flows, see [6] for Cr, r≥1, flows, see [1][11] for C0 homeomorphisms, see time one map in [6] for Cr, r≥1 diffeomorphisms. We point out it is non-hyperbolic singularity causes the degeneracy while changing one flow equivalently to another.

Variational principle for polynomial entropy on subsets of free semigroup actions

We investigate the relations between Bowen polynomial entropy on an arbitrary subset and local measure theoretic polynomial entropy of Borel probability measure for finitely generated semigroup actions. Also, we establish a variational principle for polynomial entropy on compact subsets in the context of free semigroup actions.

Slow entropy of some combinatorial constructions

Measure-theoretic slow entropy is a more refined invariant than the classical measure-theoretic entropy to characterize the complexity of dynamical systems with subexponential growth rates of distinguishable orbit types. In this paper we prove flexibility results for the values of upper and lower polynomial slow entropy of rigid transformations as well as maps admitting a good cyclic approximation. Moreover, we show that there cannot exist a general upper bound on the lower measure-theoretic slow entropy for systems of finite rank.

Horseshoes and Lyapunov exponents for Banach cocycles over non-uniformly hyperbolic systems

AbstractWe extend Katok’s result on ‘the approximation of hyperbolic measures by horseshoes’ to Banach cocycles. More precisely, let f be a $C^r(r>1)$ diffeomorphism of a compact Riemannian manifold M, preserving an ergodic hyperbolic measure $\mu $ with positive entropy, and let $\mathcal {A}$ be a Hölder continuous cocycle of bounded linear operators acting on a Banach space $\mathfrak {X}$ . We prove that there is a sequence of horseshoes for f and dominated splittings for $\mathcal {A}$ on the horseshoes, such that not only the measure theoretic entropy of f but also the Lyapunov exponents of $\mathcal {A}$ with respect to $\mu $ can be approximated by the topological entropy of f and the Lyapunov exponents of $\mathcal {A}$ on the horseshoes, respectively. As an application, we show the continuity of sub-additive topological pressure for Banach cocycles.

Variational principles for topological pressures on subsets

In this paper, we investigate the relations between various types of topological pressures and different versions of measure-theoretical pressures. We extend Feng–Huang’s variational principle for packing entropy to packing pressure and obtain two new variational principles for Pesin–Pitskel and packing pressures respectively. We show that various types of Katok pressures for an ergodic measure with respect to a potential function are equal to the sum of measure-theoretic entropy of this measure and the integral of the potential function. Moreover, we obtain Billingsley type theorem for packing pressure, which indicates that packing pressure can be determined by measure-theoretic upper local pressure of measures, and a variational principle for packing pressure of the set of generic points for any invariant ergodic Borel probability measure.

On weighted sub-additive topological and measure-theoretic pressures

Based on several types of weighted sub-additive topological pressures which arise from the theory of Carathéodory structure, we define various weighted sub-additive measure-theoretic pressures. Also, a characterization of weighted sub-additive measure-theoretic pressures for ergodic measures is derived in terms of weighted measure-theoretic entropy, which immediately indicates an inverse variational principle for weighted sub-additive topological pressure. Afterward we illustrate that the inverse variational principle can be attained on the set of generic points of the ergodic measure μ when the potential is additive.

Slow entropy for some Anosov-Katok diffeomorphisms

<p style='text-indent:20px;'>The Anosov-Katok method is one of the most powerful tools of constructing smooth volume-preserving diffeomorphisms of entropy zero with prescribed ergodic or topological properties. To measure the complexity of systems with entropy zero, invariants like slow entropy have been introduced. In this article we develop several mechanisms facilitating computation of topological and measure-theoretic slow entropy of Anosov-Katok diffeomorphisms.</p>

Formulae of volume growth for random Anosov diffeomorphisms

For $ C^2 $ random Anosov diffeomorphisms, we give a formula describing the volume growth by measure-theoretic fiber entropy and Lyapunov exponents. Furthermore, the topological fiber entropy coincides with the volume growth for $ C^2 $ fiber hyperbolic systems, which was for $ C^{\infty} $ random dynamical systems in [15] and $ C^2 $ Axiom A systems in [10]. We establish two tools, a volume lemma and a variational principle for random radii in $ C^2 $ random Anosov diffeomorphisms.

Symbolic Extensions of Amenable Group Actions and the Comparison Property

In topological dynamics, the Symbolic Extension Entropy Theorem (SEET) (Boyle and Downarowicz, 2004) describes the possibility of a lossless digitalization of a dynamical system by extending it to a subshift on finitely many symbols. The theorem gives a precise estimate on the entropy of such a symbolic extension (and hence on the necessary number of symbols). Unlike in the measure-theoretic case, where Kolmogorov–Sinai entropy serves as an estimate in an analogous problem, in the topological setup the task reaches beyond the classical theory of measure-theoretic and topological entropy. Necessary are tools from an extended theory of entropy, the theory of entropy structures developed in Downarowicz (2005). The main goal of this paper is to prove the analog of the SEET for actions of (discrete infinite) countable amenable groups: Let a countable amenable group G G act by homeomorphisms on a compact metric space X X and let M G ( X ) \mathcal {M}_{G}(X) denote the simplex of all G G -invariant Borel probability measures on X X . A function E A {E}_{\mathsf {A}} on M G ( X ) \mathcal {M}_{G}(X) equals the extension entropy function h π h^\pi of a symbolic extension π : ( Y , G ) → ( X , G ) \pi :(Y,G)\to (X,G) , where h π ( μ ) = sup { h ν ( Y , G ) : ν ∈ π − 1 ( μ ) } h^\pi (\mu )=\sup \{h_\nu (Y,G): \nu \in \pi ^{-1}(\mu )\} ( μ ∈ M G ( X ) \mu \in \mathcal {M}_{G}(X) ), if and only if E A {E}_{\mathsf {A}} is a finite affine superenvelope of the entropy structure of ( X , G ) (X,G) . Of course, the statement is preceded by the presentation of the concepts of an entropy structure and its superenvelopes, adapted from the case of Z \mathbb {Z} -actions. In full generality we are able to prove a slightly weaker version of SEET, in which symbolic extensions are replaced by quasi-symbolic extensions, i.e., extensions in form of a joining of a subshift with a zero-entropy tiling system. The notion of a tiling system is a subject of several earlier works (e.g. Downarowicz and Huczek (2018) and Downarowicz, Huczek, and Zhang (2019)) and in this paper we review and complement the theory developed there. The full version of the SEET (with genuine symbolic extensions) is proved for groups which are either residually finite or enjoy the so-called comparison property. In order to describe the range of our theorem more clearly, we devote a large portion of the paper to studying the comparison property. Our most important result in this aspect is showing that all subexponential groups have the comparison property (and thus satisfy the SEET). To summarize, the heart of the paper is the presentation of the following four major topics and the interplay between them: Symbolic extensions, Entropy structures, Tiling systems (and their encodability), The comparison property.

Sequence complexity, rigidity and logarithmic Sarnak conjecture

In this paper, we introduce sequence complexity and sequence mean complexity. It turns out that a topological dynamical system is uniformly rigid if and only if it has bounded topological sequence complexity; A measure preserving system is rigid if and only if it has bounded measure theoretic sequence complexity if and only if it has bounded measure theoretic sequence mean complexity. We obtain a formula of measure-theoretic entropy of an ergodic measure preserving system. We also prove that a topological dynamical systems $ (X, T) $ meets the Logarithmic Sarnak conjecture if each invariant measure of $ (X, T) $ has sub-linear measure theoretic sequence mean complexity.

Relative topological conditional entropy and a Ledrappier's type variational principle for it

In this paper we provide a sufficient condition for the existence of invariant measures with maximal relative measure-theoretic entropy, by introducing a new invariant for any factor map between topological dynamical systems, the concept of relative topological conditional entropy. It is proved a Ledrappier's type variational principle concerning relative topological conditional entropy. Consequently, if the factor map has zero relative topological conditional entropy (such a factor map is called asymptotically $ h $-expansive), then there exist invariant measures with maximal relative measure-theoretic entropy (i.e., whose relative measure-theoretic entropy equals exactly relative topological conditional entropy of the factor map). We explore further properties of relative topological conditional entropy, for example, the addition formula for a product factor map, the estimation of it for composition of two factor maps, and so on. We also interpret relative topological conditional entropy, respectively, using relative topological entropy of subsets and Bowen's dimensional entropy.

On the induced measure-theoretic entropy for random dynamical systems

In this paper, we introduce the concepts of induced topological entropy and induced measure-theoretic entropy for random dynamical systems. By providing the topological version of induced measure-theoretic entropy, we establish the Katok entropy formula for a continuous bundle random dynamical system over an invertible ergodic system. We also consider the induced version with lower and upper pointwise dimensions, and establish a measure-theoretic entropy formula for lower and upper induced pointwise dimensions for a continuous bundle random dynamical system over an invertible ergodic system.