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In this article, we combine the perspectives of density, entropy, andmultifractal analysis to investigate the structure of ergodic measures. Weprove that for each transitive topologically Anosov system $(X,f)$, eachcontinuous 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 ageneral framework and use it to obtain intermediate entropy property of ergodicmeasures with same Lyapunov exponent for non-hyperbolic step skew-products,elliptic $\operatorname{SL}(2, \mathbb{R})$ cocycles and robustlynon-hyperbolic transitive diffeomorphisms. Moreover, we get generalized resultson multiple functions and use them to obtain the intermediate Hausdorffdimension of ergodic measures for transitive average conformal orquasi-conformal Anosov diffeomorphisms, that is $\left\{\operatorname{dim}_H\mu: \mu\in M_f^e(M)\right\}= \left\{\operatorname{dim}_H \mu: \mu\inM_f(M)\right\}.$ In this process, we introduce and establish a'multi-horseshoe' entropy-dense property and use it to get the goal combinedwith the well-known conditional variational principles. As applications, wealso obtain many new observations on various other quantitative spectrumsincluding Lyapunov exponents, first return rate, geometric pressure, unstableHausdorff dimension, etc.

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