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 $\Lambda$ of a flow $\Phi$ and two continuous function $g,$ $h$ on $\Lambda,$ we obtain $$\mathrm{Int}\left\{h_{\mu}(\Phi):\mu\in \mathcal{M}_{erg}(\Phi,\Lambda)\text{ and }\int g d\mu=\alpha\right\}=\mathrm{Int}\left\{h_{\mu}(\Phi):\mu\in \mathcal{M}(\Phi,\Lambda) \text{ and }\int g d\mu=\alpha\right\},$$ $$\mathrm{Int}\left\{\int g d\mu:\mu\in \mathcal{M}_{erg}(\Phi,\Lambda)\text{ and }h_{\mu}(\Phi)=c\right\}=\mathrm{Int}\left\{\int g d\mu:\mu\in \mathcal{M}(\Phi,\Lambda) \text{ and }h_{\mu}(\Phi)=c\right\}$$ and $$\mathrm{Int}\left\{\int h d\mu:\mu\in \mathcal{M}_{erg}(\Phi,\Lambda)\text{ and }\int g d\mu=\alpha\right\}=\mathrm{Int}\left\{\int h d\mu:\mu\in \mathcal{M}(\Phi,\Lambda) \text{ and }\int g d\mu=\alpha\right\}$$ for any $\alpha\in \left(\inf_{\mu\in \in \mathcal{M}(\Phi,\Lambda) }\int g d\mu, \, \sup_{\mu\in \in \mathcal{M}(\Phi,\Lambda) }\int g d\mu\right)$ 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.