Smale马蹄的丰富性与遍历测度：基于多重分形分析与各种定量谱的研究
Abundance of Smale's horseshoes and ergodic measures via multifractal analysis and various quantitative spectrums
AI总结：
本文结合密度、熵和多重分形分析研究遍历测度结构，证明了传递Anosov系统中遍历测度的稠密$G_δ$性质，建立了“多马蹄”熵稠密性质的一般框架，并推广至非双曲系统与多种定量谱分析。
摘要：
在本文中，我们结合密度、熵和多重分形分析的视角来研究遍历测度的结构。我们证明，对于每个传递的拓扑Anosov系统$(X,f)$、$X$上的每个连续函数$φ$以及每个$(a,h)\in \mathrm{Int}\{(\int φdμ, h_μ(f)):μ\in M_f(X)\}$，集合$\{μ\in M_f^e(X): (\int φdμ, h_μ(f))=(a,h)\}$是非空的，并且包含$\{μ\in M_f(X): (\int φdμ, h_μ(f))=(a,h)\}$的一个稠密$G_δ$子集。同时，结合非双曲系统和环面的发展，我们给出了一个一般性框架，并利用它获得了非双曲阶梯斜积、椭圆$\operatorname{SL}(2, \mathbb{R})$环面和鲁棒非双曲传递微分同胚的具有相同Lyapunov指数的遍历测度的中间熵性质。此外，我们得到了关于多个函数的推广结果，并利用它们获得了传递平均共形或拟共形Anosov微分同胚的遍历测度的中间Hausdorff维数，即$\left\{\operatorname{dim}_H μ: μ\in M_f^e(M)\right\}= \left\{\operatorname{dim}_H μ: μ\in M_f(M)\right\}$。在此过程中，我们引入并建立了“多马蹄”熵稠密性质，并结合众所周知的条件变分原理来实现目标。作为应用，我们还获得了关于各种其他定量谱的许多新观察，包括Lyapunov指数、首次返回率、几何压力、不稳定Hausdorff维数等。
英文摘要：
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 $φ$ on $X$ and each $(a,h)\in \mathrm{Int}\{(\int φdμ, h_μ(f)):μ\in M_f(X)\},$ the set $\{μ\in M_f^e(X): (\int φdμ, h_μ(f))=(a,h)\}$ is non-empty and contains a dense $G_δ$ subset of $\{μ\in M_f(X): (\int φdμ, h_μ(f))=(a,h)\}.$ 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.
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