Conditional intermediate entropy and Birkhoff average properties of hyperbolic flows

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 $\mathrm{\Lambda }$ of a flow $\mathrm{\Phi }$ and two continuous function $g,$ $h$ on $\mathrm{\Lambda },$ we obtain $$\mathrm{I}\mathrm{n}\mathrm{t}\{h_{\mu }(\mathrm{\Phi }):\mu \in {\mathcal{M}}_{erg}(\mathrm{\Phi },\mathrm{\Lambda })\text{ and }\int gd\mu =\alpha \}=\mathrm{I}\mathrm{n}\mathrm{t}\{h_{\mu }(\mathrm{\Phi }):\mu \in \mathcal{M}(\mathrm{\Phi },\mathrm{\Lambda })\text{ and }\int gd\mu =\alpha \},$$ $$\mathrm{I}\mathrm{n}\mathrm{t}\{\int gd\mu :\mu \in {\mathcal{M}}_{erg}(\mathrm{\Phi },\mathrm{\Lambda })\text{ and }{h}_{\mu }(\mathrm{\Phi })=c\}=\mathrm{I}\mathrm{n}\mathrm{t}\{\int gd\mu :\mu \in \mathcal{M}(\mathrm{\Phi },\mathrm{\Lambda })\text{ and }{h}_{\mu }(\mathrm{\Phi })=c\}$$ and $$\mathrm{I}\mathrm{n}\mathrm{t}\{\int hd\mu :\mu \in {\mathcal{M}}_{erg}(\mathrm{\Phi },\mathrm{\Lambda })\text{ and }\int gd\mu =\alpha \}=\mathrm{I}\mathrm{n}\mathrm{t}\{\int hd\mu :\mu \in \mathcal{M}(\mathrm{\Phi },\mathrm{\Lambda })\text{ and }\int gd\mu =\alpha \}$$ for any $\alpha \in ({\inf }_{\mu \in \in \mathcal{M}(\mathrm{\Phi },\mathrm{\Lambda })}\int gd\mu ,{\sup }_{\mu \in \in \mathcal{M}(\mathrm{\Phi },\mathrm{\Lambda })}\int gd\mu )$ and any $c\in (0,h_{top}(\mathrm{\Lambda }))\mathrm{.}$ 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.