Conditional intermediate entropy and Birkhoff average properties of hyperbolic flows
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2022
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| _version_ | 1866929596872523776 |
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| author | Hou, Xiaobo Tian, Xueting |
| author_facet | Hou, Xiaobo Tian, Xueting |
| contents | 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 $μ$ of $f$ satisfying $h_μ(f)=c$. In this paper we consider a conditional intermediate metric entropy property and two conditional intermediate Birkhoff average properties for flows. For a basic set $Λ$ of a flow $Φ$ and two continuous function $g,$ $h$ on $Λ,$ we obtain $$\mathrm{Int}\left\{h_μ(Φ):μ\in \mathcal{M}_{erg}(Φ,Λ)\text{ and }\int g dμ=α\right\}=\mathrm{Int}\left\{h_μ(Φ):μ\in \mathcal{M}(Φ,Λ) \text{ and }\int g dμ=α\right\},$$ $$\mathrm{Int}\left\{\int g dμ:μ\in \mathcal{M}_{erg}(Φ,Λ)\text{ and }h_μ(Φ)=c\right\}=\mathrm{Int}\left\{\int g dμ:μ\in \mathcal{M}(Φ,Λ) \text{ and }h_μ(Φ)=c\right\}$$ and $$\mathrm{Int}\left\{\int h dμ:μ\in \mathcal{M}_{erg}(Φ,Λ)\text{ and }\int g dμ=α\right\}=\mathrm{Int}\left\{\int h dμ:μ\in \mathcal{M}(Φ,Λ) \text{ and }\int g dμ=α\right\}$$ for any $α\in \left(\inf_{μ\in \in \mathcal{M}(Φ,Λ) }\int g dμ, \, \sup_{μ\in \in \mathcal{M}(Φ,Λ) }\int g dμ\right)$ and any $c\in (0,h_{top}(Λ)).$ 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2209_02959 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | Conditional intermediate entropy and Birkhoff average properties of hyperbolic flows Hou, Xiaobo Tian, Xueting Dynamical Systems 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 $μ$ of $f$ satisfying $h_μ(f)=c$. In this paper we consider a conditional intermediate metric entropy property and two conditional intermediate Birkhoff average properties for flows. For a basic set $Λ$ of a flow $Φ$ and two continuous function $g,$ $h$ on $Λ,$ we obtain $$\mathrm{Int}\left\{h_μ(Φ):μ\in \mathcal{M}_{erg}(Φ,Λ)\text{ and }\int g dμ=α\right\}=\mathrm{Int}\left\{h_μ(Φ):μ\in \mathcal{M}(Φ,Λ) \text{ and }\int g dμ=α\right\},$$ $$\mathrm{Int}\left\{\int g dμ:μ\in \mathcal{M}_{erg}(Φ,Λ)\text{ and }h_μ(Φ)=c\right\}=\mathrm{Int}\left\{\int g dμ:μ\in \mathcal{M}(Φ,Λ) \text{ and }h_μ(Φ)=c\right\}$$ and $$\mathrm{Int}\left\{\int h dμ:μ\in \mathcal{M}_{erg}(Φ,Λ)\text{ and }\int g dμ=α\right\}=\mathrm{Int}\left\{\int h dμ:μ\in \mathcal{M}(Φ,Λ) \text{ and }\int g dμ=α\right\}$$ for any $α\in \left(\inf_{μ\in \in \mathcal{M}(Φ,Λ) }\int g dμ, \, \sup_{μ\in \in \mathcal{M}(Φ,Λ) }\int g dμ\right)$ and any $c\in (0,h_{top}(Λ)).$ 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. |
| title | Conditional intermediate entropy and Birkhoff average properties of hyperbolic flows |
| topic | Dynamical Systems |
| url | https://arxiv.org/abs/2209.02959 |