Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Abundance of Smale's horseshoes and ergodic measures via multifractal analysis and various quantitative spectrums

Yiwei Dong, Xiaobo Hou, Xueting Tian

Abstract

In this article, we combine the perspectives of density, entropy, and multifractal analysis to investigate the structure of ergodic measures. We prove that for each transitive topologically Anosov system $(X,f)$, each continuous function $\varphi$ on $X$ and each $(a,h)\in \mathrm{Int}\{(\int \varphi dμ, h_μ(f)):μ\in M_f(X)\},$ the set $\{μ\in M_f^e(X): (\int \varphi dμ, h_μ(f))=(a,h)\}$ is non-empty and contains a dense $G_δ$ subset of $\{μ\in M_f(X): (\int \varphi dμ, h_μ(f))=(a,h)\}.$ Meanwhile, combining the development of non-hyperbolic systems and cocycles we give a general framework and use it to obtain intermediate entropy property of ergodic measures with same Lyapunov exponent for non-hyperbolic step skew-products, elliptic $\operatorname{SL}(2, \mathbb{R})$ cocycles and robustly non-hyperbolic transitive diffeomorphisms. Moreover, we get generalized results on multiple functions and use them to obtain the intermediate Hausdorff dimension of ergodic measures for transitive average conformal or quasi-conformal Anosov diffeomorphisms, that is $\left\{\operatorname{dim}_H μ: μ\in M_f^e(M)\right\}= \left\{\operatorname{dim}_H μ: μ\in M_f(M)\right\}.$ In this process, we introduce and establish a 'multi-horseshoe' entropy-dense property and use it to get the goal combined with the well-known conditional variational principles. As applications, we also obtain many new observations on various other quantitative spectrums including Lyapunov exponents, first return rate, geometric pressure, unstable Hausdorff dimension, etc.

Abundance of Smale's horseshoes and ergodic measures via multifractal analysis and various quantitative spectrums

Abstract

In this article, we combine the perspectives of density, entropy, and multifractal analysis to investigate the structure of ergodic measures. We prove that for each transitive topologically Anosov system , each continuous function on and each the set is non-empty and contains a dense subset of Meanwhile, combining the development of non-hyperbolic systems and cocycles we give a general framework and use it to obtain intermediate entropy property of ergodic measures with same Lyapunov exponent for non-hyperbolic step skew-products, elliptic cocycles and robustly non-hyperbolic transitive diffeomorphisms. Moreover, we get generalized results on multiple functions and use them to obtain the intermediate Hausdorff dimension of ergodic measures for transitive average conformal or quasi-conformal Anosov diffeomorphisms, that is In this process, we introduce and establish a 'multi-horseshoe' entropy-dense property and use it to get the goal combined with the well-known conditional variational principles. As applications, we also obtain many new observations on various other quantitative spectrums including Lyapunov exponents, first return rate, geometric pressure, unstable Hausdorff dimension, etc.
Paper Structure (54 sections, 75 theorems, 160 equations, 5 figures)

This paper contains 54 sections, 75 theorems, 160 equations, 5 figures.

Table of Contents

  1. Introduction
  2. Intermediate entropy property of ergodic measures with same level
  3. Topologically Anosov systems
  4. A general framework
  5. Non-hyperbolic step skew-products
  6. Elliptic $\operatorname{SL}(2, \mathbb{R})$ cocycles
  7. Robustly non-hyperbolic transitive diffeomorphisms
  8. 'Multi-horseshoe' entropy-dense property
  9. Intermediate Hausdorff dimension of ergodic measures
  10. Lyapunov spectrum
  11. First return rate of ergodic measures
  12. Other applications
  13. Intermediate geometric pressure of ergodic measures
  14. Intermediate unstable Hausdorff dimension of ergodic measures
  15. Intermediate Hausdorff dimension of generic points of ergodic measures
  16. ...and 39 more sections

Key Result

Theorem 1.1

BarreiraSaussol2001 Suppose $(X, f)$ is a dynamical system whose entropy function $\mathcal{E}_f$ is upper semi-continuous. Given $\varphi\in C(X).$ If $b_1\varphi+b_2$ has a unique equilibrium measure for any $b_1,b_2\in\mathbb{R},$ then for any $a\in \mathrm{Int}(\mathcal{P}_\varphi(\mathcal{M}_f

Figures (5)

  • Figure 1: Entropy and multifractal analysis
  • Figure 2: Graph of $(\int \varphi d\mu, h_\mu(f))$
  • Figure 3: Graph of $(\chi(\mu), h_\mu(F))$
  • Figure 4: Graph of $(\int \varphi d\mu, \dim_H\mu)$
  • Figure 5: Relationships between the main statements

Theorems & Definitions (127)

  • Theorem 1.1
  • Definition 1.2
  • Theorem A
  • Remark 1.3
  • Remark 1.4
  • Remark 1.5
  • Corollary A
  • Theorem B
  • Remark 1.6
  • Remark 1.7
  • ...and 117 more