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 $(X,f)$, each continuous function $\varphi$ on $X$ and each $(a,h)\in \mathrm{Int}\{(\int \varphi d\mu, h_\mu(f)):\mu\in M_f(X)\},$ the set $\{\mu\in M_f^e(X): (\int \varphi d\mu, h_\mu(f))=(a,h)\}$ is non-empty and contains a dense $G_\delta$ subset of $\{\mu\in M_f(X): (\int \varphi d\mu, h_\mu(f))=(a,h)\}.$ 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 $\left\{\operatorname{dim}_H \mu: \mu\in M_f^e(M)\right\}= \left\{\operatorname{dim}_H \mu: \mu\in M_f(M)\right\}.$ 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.