论文标题
通过多重分析和各种定量谱系
Abundance of Smale's horseshoes and ergodic measures via multifractal analysis and various quantitative spectrums
论文作者
论文摘要
在本文中，我们结合了密度，熵和多重分析分析的观点，以研究千古措施的结构。我们证明，对于拓扑上的每个传递性，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； （\intφdμ，h_μ（f））=（a，h）\} $是非空的，并且包含一个密集的$g_Δ$子集$ \ {μ\ in m_f（x）中的$ \ {μ\（x）：（\intφdμ，h__μ（f））=（a，h）=（a，h）\}。框架并使用它来获得具有相同lyapunov指数的Ergodic措施的中间熵属性，用于非毛细血管步骤偏斜产品，椭圆$ \ permatatorname {sl}（2，\ Mathbb {r}）$ cocycles $ cocycles $ cocycles $ botally-Hy-Hyperbolic Enstrical Exprantitive Diffefeomorphismssssssssssss。此外，我们在多个功能上获得广泛的结果，并使用它们来获取中间的Hausdorff尺寸，用于传递的平均共形或准式形式的Anosov anosov diffefemorlisms，即$ \ weft \ left \ weft \ {\ operatoRatorname {dim}} _h} _hμ：：μ：μ：m__f^e（m） \ left \ {\ operatorname {dim}_hμ：μ\ in m_f（m）\ right \}。$在此过程中，我们介绍并建立了“多用马shoe”熵密度的属性，并使用它来使目标与众所周知的条件各种差异原理结合在一起。作为应用程序，我们还获得了许多其他定量频谱的新观察结果，包括Lyapunov指数，第一返回率，几何压力，不稳定的Hausdorff Dimension等。
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.
