Katok's Entorpy Formula of Unstable Metric Entropy for Partially Hyperbolic Diffeomorphisms

In this paper, we define and study unstable measure theoretic pressure for C 1 -smooth partially hyperbolic diffeomorphisms with sub-additive potentials. We show that this measure theoretic pressure for any ergodic measure equals the corresponding unstable measure theoretic entropy plus the Lyapunov exponent of the potentials with respect to the ergodic measure. On the other hand, we also give other definitions of unstable metric pressure, in terms of Bowen's picture and the capacity picture. We show that all definitions of unstable metric pressure, including the one defined at the beginning, actually coincide for any ergodic measure.

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“…The parameters c 1 , c 2 above are not essential, only for technical applications later. Such constructions appeared before in [2]. Lemma 3.5.…”

Section: The Case Of Additive Potentialsmentioning

confidence: 87%

Unstable metric pressure of partially hyperbolic diffeomorphisms with sub-additive potentials

Zhang

Zhou

2020

Nonlinearity

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“…As a generalization of unstable topological entropy, unstable pressure is defined in [2], where the so-called u-equilibrium states are introduced and investigated finely. Recently, a version of local variational principle and Katok's entropy formula for unstable metric entropy are given in [12] and [3] respectively.…”

Section: Introductionmentioning

confidence: 99%

Unstable entropy and unstable pressure for partially hyperbolic endomorphisms

Wang

Zhu

2020

Journal of Mathematical Analysis and Applications

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In this paper, unstable metric entropy, unstable topological entropy and unstable pressure for partially hyperbolic endomorphisms are introduced and investigated. A version of Shannon-McMillan-Breiman Theorem is established, and a variational principle is formulated, which gives a relationship between unstable metric entropy and unstable pressure (unstable topological entropy). As an application of the variational principle, some results on the uequilibrium states are given.

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Katok's Entorpy Formula of Unstable Metric Entropy for Partially Hyperbolic Diffeomorphisms

Abstract: This paper is devoted to establishing the Katok's entropy formula of unstable metric entropy which is the entropy caused by the unstable part of partially hyperbolic systems.

Cited by 2 publications

References 14 publications

Unstable metric pressure of partially hyperbolic diffeomorphisms with sub-additive potentials

Unstable metric pressure of partially hyperbolic diffeomorphisms with sub-additive potentials

Unstable entropy and unstable pressure for partially hyperbolic endomorphisms

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