Conditional intermediate entropy and Birkhoff average properties of hyperbolic flows

Abstract

Katok conjectured that every $C^2$ diffeomorphism $f$ on a Riemannian manifold has the intermediate entropy property, that is, for any constant $c\in [0,h_{top}(f))$, there exists an ergodic measure $\mu$ of $f$ satisfying $h_{\mu }(f)=c$. In this paper we consider a conditional intermediate metric entropy property and two conditional intermediate Birkhoff average properties for flows. For a basic set $\Lambda$ of a flow $\Phi$ and two continuous function $g,$ $h$ on $\Lambda ,$ we obtain $$\mathrm{Int}\left\{h_{\mu }(\Phi ):\mu \in {\mathcal{M}}_{erg}(\Phi ,\Lambda )\text{ and }\int gd\mu =\alpha \right\}=\mathrm{Int}\left\{h_{\mu }(\Phi ):\mu \in \mathcal{M}(\Phi ,\Lambda )\text{ and }\int gd\mu =\alpha \right\},$$ $$\mathrm{Int}\left\{\int gd\mu :\mu \in {\mathcal{M}}_{erg}(\Phi ,\Lambda )\text{ and }{h}_{\mu }(\Phi )=c\right\}=\mathrm{Int}\left\{\int gd\mu :\mu \in \mathcal{M}(\Phi ,\Lambda )\text{ and }{h}_{\mu }(\Phi )=c\right\}$$ and $$\mathrm{Int}\left\{\int hd\mu :\mu \in {\mathcal{M}}_{erg}(\Phi ,\Lambda )\text{ and }\int gd\mu =\alpha \right\}=\mathrm{Int}\left\{\int hd\mu :\mu \in \mathcal{M}(\Phi ,\Lambda )\text{ and }\int gd\mu =\alpha \right\}$$ for any $\alpha \in \left({\inf }_{\mu \in \in \mathcal{M}(\Phi ,\Lambda )}\int gd\mu ,{\sup }_{\mu \in \in \mathcal{M}(\Phi ,\Lambda )}\int gd\mu \right)$ and any $c\in (0,h_{top}(\Lambda )).$ In this process, we establish 'multi-horseshoe' entropy-dense property and use it to get the goal combined with conditional variational principles. We also obtain same result for singular hyperbolic attractors.

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