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Smooth surface systems may contain smooth curves which have no measure of maximal entropy

Xulei Wang, Guohua Zhang

TL;DR

The paper reveals intricate behavior of measures of maximal entropy for analytic subsets in dynamical systems, showing that smoothness alone does not guarantee MMEs for subsets and that analyticity induces a multiplicity of MMEs when such measures exist. It introduces and exploits the concepts of packing and Bowen entropies, gauge-function dimensions, and increasing countable slices to characterize when an analytic set carries MMEs. The main results establish dichotomies and equivalences: in $h$-expansive or asymptotically $h$-expansive systems, analytic $Z$ with positive entropy either has MMEs or can be decomposed into pieces with strictly smaller entropy, and under suitable conditions, Bowen MMEs can be characterized via gauge-function measures. A central construction---a smooth skew-product example on $[0,1]^2$ with a diagonal curve lacking MMEs---demonstrates that positive entropy does not ensure MMEs for analytic subsets, underscoring the nuanced distinction between global invariant measures and subset entropies. The work also links dimension-theoretic properties (Hausdorff/dimension) to entropy-maximizing measures, broadening the understanding of entropy in analytic-set dynamics and providing tools for further exploration of MMEs in complex systems.

Abstract

In this paper, we study Borel probability measures of maximal entropy for analytic subsets in a dynamical system. It is well known that higher smoothness of the map over smooth space plays important role in the study of invariant measures of maximal entropy. A famous theorem of Newhouse states that smooth diffeomorphisms on compact manifolds without boundary have invariant measures of maximal entropy. However, we show that the situation becomes completely different when we study measures of maximal entropy for analytic subsets. Namely, we construct a smooth surface system which contains a smooth curve having no Borel probability measure of maximal entropy. Another evidence to show this difference is that, once an analytic set has one measure of maximal entropy, then the set has many measures of maximal entropy (no matter if we consider packing or Bowen entropy). For a general dynamical system with positive entropy $h_\mathrm{top}(T)$, we shall show that the system contains not only a Borel subset which has Borel probability measures of maximal entropy and has entropy sufficiently close to $h_\mathrm{top}(T)$, but also a Borel subset which has no Borel probability measures of maximal entropy and has entropy equal to the arbitrarily given positive real number which is at most $h_\mathrm{top}(T)$. We also provide in all $h$-expansive systems a full characterization for analytic subsets which have Borel probability measures of maximal entropy. Consequently, if let $Z\subset \mathbb{R}^n$ be any analytic subset with positive Hausdorff dimension in Euclidean space, then the set $Z$ either has a measure of full lower Hausdorff dimension, or it can be partitioned into a union of countably many analytic sets $\{Z_i\}_{i\in \mathbb{N}}$ with $\dim_{\mathcal{H}}(Z_i) < \dim_{\mathcal{H}}(Z)$ for each $i$.

Smooth surface systems may contain smooth curves which have no measure of maximal entropy

TL;DR

The paper reveals intricate behavior of measures of maximal entropy for analytic subsets in dynamical systems, showing that smoothness alone does not guarantee MMEs for subsets and that analyticity induces a multiplicity of MMEs when such measures exist. It introduces and exploits the concepts of packing and Bowen entropies, gauge-function dimensions, and increasing countable slices to characterize when an analytic set carries MMEs. The main results establish dichotomies and equivalences: in -expansive or asymptotically -expansive systems, analytic with positive entropy either has MMEs or can be decomposed into pieces with strictly smaller entropy, and under suitable conditions, Bowen MMEs can be characterized via gauge-function measures. A central construction---a smooth skew-product example on with a diagonal curve lacking MMEs---demonstrates that positive entropy does not ensure MMEs for analytic subsets, underscoring the nuanced distinction between global invariant measures and subset entropies. The work also links dimension-theoretic properties (Hausdorff/dimension) to entropy-maximizing measures, broadening the understanding of entropy in analytic-set dynamics and providing tools for further exploration of MMEs in complex systems.

Abstract

In this paper, we study Borel probability measures of maximal entropy for analytic subsets in a dynamical system. It is well known that higher smoothness of the map over smooth space plays important role in the study of invariant measures of maximal entropy. A famous theorem of Newhouse states that smooth diffeomorphisms on compact manifolds without boundary have invariant measures of maximal entropy. However, we show that the situation becomes completely different when we study measures of maximal entropy for analytic subsets. Namely, we construct a smooth surface system which contains a smooth curve having no Borel probability measure of maximal entropy. Another evidence to show this difference is that, once an analytic set has one measure of maximal entropy, then the set has many measures of maximal entropy (no matter if we consider packing or Bowen entropy). For a general dynamical system with positive entropy , we shall show that the system contains not only a Borel subset which has Borel probability measures of maximal entropy and has entropy sufficiently close to , but also a Borel subset which has no Borel probability measures of maximal entropy and has entropy equal to the arbitrarily given positive real number which is at most . We also provide in all -expansive systems a full characterization for analytic subsets which have Borel probability measures of maximal entropy. Consequently, if let be any analytic subset with positive Hausdorff dimension in Euclidean space, then the set either has a measure of full lower Hausdorff dimension, or it can be partitioned into a union of countably many analytic sets with for each .
Paper Structure (18 sections, 32 theorems, 161 equations, 2 figures)

This paper contains 18 sections, 32 theorems, 161 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Preliminariles
  3. Packing topological entropy
  4. Hausdorff dimension and Bowen topological entropy
  5. Measure-theoretical upper and lower entropies
  6. $h$-expansive system and asymptotically $h$-expansive system
  7. Basic properties concerning entropy of analytic sets
  8. Proof of easier parts of our results
  9. Construction of Example \ref{['submanifold counterexample']}
  10. Proof of Theorem \ref{['MME Packing h-expan']}
  11. Proof of Theorem \ref{['MME Bowen']} for compact subsets
  12. Proof of $(3) \Longrightarrow (1)$ of Theorem \ref{['MME Bowen-s6']}
  13. Equivalent formulation of $\mathcal{M}^{b}(Z)$
  14. Proof of $(3) \Longrightarrow (1)$ of Theorem \ref{['MME Bowen-s6']}
  15. Proof of $(2) \Longrightarrow (3)$ of Theorem \ref{['MME Bowen-s6']} for $h$-expansive systems
  16. ...and 3 more sections

Key Result

Proposition 1.3

Let $Y\subset Z$ both be analytic sets in $X$, and $\mu\in\mathcal{M}(X)$ with $\mu(Z)=1$ and $\mu (Y) > 0$. Let $\nu$ be the normalized measure of $\mu$ restricted onto $Y$.

Figures (2)

  • Figure 1: Function of $a(x)$
  • Figure 2: Diagonal set

Theorems & Definitions (70)

  • Definition 1.1
  • Example 1.2
  • Proposition 1.3
  • Theorem 1.4
  • Definition 1.5
  • Theorem 1.6
  • Definition 1.7
  • Theorem 1.8
  • Corollary 1.9
  • Corollary 1.10
  • ...and 60 more