论文标题
双曲线流的条件中间熵和伯克霍夫的平均特性
Conditional intermediate entropy and Birkhoff average properties of hyperbolic flows
论文作者
论文摘要
卡托克（Katok）猜想，每一个$ c^{2} $ diffeomormormormormormormorphism $ f $ f $ f $ in Riemannian歧管具有中间熵属性，也就是说，对于[0，h_ {top}（f））$中的任何常数$ c \ c \ in [0，h_ {top}（f））$，都有一个e g $ f $ $ f $ f $ $ f $ f $满足$ h_ h__ h__ f）= c $ f $ f $ f）。在本文中，我们考虑了有条件的中间度量熵特性和两个条件中间的伯克霍夫平均水平。对于$φ$的基本集合$λ$和两个连续函数$ g，$λ上的$ $ h $，$，我们获得$$ \ mathrm {int} \ left \ left \ {h_μ（φ） dμ=α\ right \} = \ mathrm {int} \ left \ {h_μ（φ）：μ\ in \ mathcal {m} {m}（φ，λ）\ text {and} \ text {and} \ int gdμ=α=α\ right \ right \ right \}，$ $ $ $ $ $ $ $ $ \ math \ w weft f lew tewsim; \ Mathcal {M} _ {erg}（φ，λ）\ text {and}h_μ（φ）= c \ right \} = \ m righrm {int} \ left \ left \ left \ weft \ weled {\ int gdμ：μ\ }h_μ（φ）= c \ right \} $$和$$ \ mathrm {int} \ left \ {\ int hdμ：μ\ in \ mathcal {m} _ {erg} _ {erg} _ {erg}（φ，λ） dμ=α\ right \} = \ mathrm {int} \ left \ {\ int Hdμ：μ\ in \ Mathcal {m} {m}（φ，λ）\ text {and} \ int Gdμ=α=α\ right \ right \ right \ right \} \ Mathcal {m}（φ，λ）} \ int gdμ，\，\ sup_ {μ\ in \ in \ in \ Mathcal {m}（m}（φ，λ）} \ int gdμ\ right）$和任何$ c \ in（0，h_ top}（top}（λ）（λ））。使用它将目标与有条件的变异原则结合在一起。我们也获得了奇异双曲线吸引子的相同结果。
Katok conjectured that every $C^{2}$ diffeomorphism $f$ on a Riemannian manifold has the intermediate entropy property, that is, for any constant $c \in[0, h_{top}(f))$, there exists an ergodic measure $μ$ of $f$ satisfying $h_μ(f)=c$. In this paper we consider a conditional intermediate metric entropy property and two conditional intermediate Birkhoff average properties for flows. For a basic set $Λ$ of a flow $Φ$ and two continuous function $g,$ $h$ on $Λ,$ we obtain $$\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.
