Conditional intermediate entropy and Birkhoff average properties of hyperbolic flows

TL;DR

This paper investigates conditional entropy and Birkhoff average properties of hyperbolic flows, extending Katok's conjecture, and establishes density and variational principles for measures with prescribed properties.

Contribution

It introduces conditional intermediate entropy and Birkhoff average properties for hyperbolic flows, proving their density and establishing connections with variational principles.

Findings

Established equality of interior sets of entropy for ergodic and general measures.

Proved the density of entropy values in specified intervals.

Extended results to singular hyperbolic attractors.

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 $$\mathrm{Int}\left\{h_{\mu}(\Phi):\mu\in \mathcal{M}_{erg}(\Phi,\Lambda)\text{ and }\int g d\mu=\alpha\right\}=\mathrm{Int}\left\{h_{\mu}(\Phi):\mu\in \mathcal{M}(\Phi,\Lambda) \text{ and }\int g d\mu=\alpha\right\},$$ $$\mathrm{Int}\left\{\int g d\mu:\mu\in \mathcal{M}_{erg}(\Phi,\Lambda)\text{ and }h_{\mu}(\Phi)=c\right\}=\mathrm{Int}\left\{\int g d\mu:\mu\in \mathcal{M}(\Phi,\Lambda) \text{ and }h_{\mu}(\Phi)=c\right\}$$ 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.