Multi-horseshoe dense property and intermediate entropy property of ergodic measures with same level

Katok conjectured that for every $C^{2}$ diffeomorphism $f$ on a Riemannian manifold $X$, the set $\{h_{\mu}(f):\mu \text{ is an ergodic measure for } (X,f)\}$ includes $[0, h_{top}(f))$. In this paper we obtained a refined Katok's conjecture on intermediate metric entropies of ergodic measures with same level that for a transitive locally maximal hyperbolic set or a transitive two-side subshit of finite type, one has $$\mathrm{Int}(\{h_{\mu}(f):\mu\in M_{erg}(f,X) \text{ and }\int\varphi d\mu=a\})=\mathrm{Int}(\{h_{\mu}(f):\mu\in M(f,X) \text{ and }\int\varphi d\mu=a\})$$ for any $a\in \left(\inf_{\mu\in M(f,X)}\int\varphi d\mu, \, \sup_{\mu\in M(f,X)}\int\varphi d\mu\right)$ and any continuous function $\varphi$. In this process, we establish 'multi-horseshoe' entropy-dense property and use it to get the goal combined with conditional variational principles