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.