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Conditional intermediate entropy and Birkhoff average properties of hyperbolic flows

Part of: Dynamical systems with hyperbolic behavior Topological dynamics Ergodic theory Smooth dynamical systems: general theory

Published online by Cambridge University Press:  14 November 2023

XIAOBO HOU Open the ORCID record for XIAOBO HOU [Opens in a new window]  and
XUETING TIAN
XIAOBO HOU
Affiliation:
School of Mathematical Science, Fudan University, Shanghai 200433, PR China (e-mail: 20110180003@fudan.edu.cn)
XUETING TIAN*
Affiliation:
School of Mathematical Science, Fudan University, Shanghai 200433, PR China (e-mail: 20110180003@fudan.edu.cn)
*
e-mail: xuetingtian@fudan.edu.cn
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Abstract

Katok [Lyapunov exponents, entropy and periodic points of diffeomorphisms. Publ. Math. Inst. Hautes Études Sci. 51 (1980), 137–173] conjectured that every $C^{2}$ diffeomorphism f on a Riemannian manifold has the intermediate entropy property, that is, for any constant $c \in [0, h_{\mathrm {top}}(f))$, there exists an ergodic measure $\mu $ of f satisfying $h_{\mu }(f)=c$. In this paper, we obtain a conditional intermediate metric entropy property and two conditional intermediate Birkhoff average properties for basic sets of flows that characterize the refined roles of ergodic measures in the invariant ones. In this process, we establish a ‘multi-horseshoe’ entropy-dense property and use it to get the goal combined with conditional variational principles. We also obtain the same result for singular hyperbolic attractors.

Keywords

hyperbolic setssingular hyperbolic attractormetric entropyhorseshoesymbolic dynamics

MSC classification

Primary: 37A25: Ergodicity, mixing, rates of mixing 37C10: Vector fields, flows, ordinary differential equations
Secondary: 37D20: Uniformly hyperbolic systems (expanding, Anosov, Axiom A, etc.) 37C70: Attractors and repellers, topological structure 37B10: Symbolic dynamics

Information

Type
Original Article
Information
Ergodic Theory and Dynamical Systems , Volume 44 , Issue 8 , August 2024 , pp. 2257 - 2288
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Copyright
© The Author(s), 2023. Published by Cambridge University Press

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