Unstable topological entropy in mean u-metrics for partially hyperbolic systems

Abstract

In this paper, we give the definition of unstable topological entropy in mean u-metrics (the mean metrics in the unstable manifold) for partially hyperbolic systems. By establishing Katok's entropy formula of unstable metric entropy in mean u-metrics, we prove that the new unstable topological entropy is equal to the unstable topological entropy defined in Bowen u-metrics (the Bowen metrics in the unstable manifold). Finally, we obtain the variational principle related to the unstable topological entropy defined in mean u-metrics and unstable metric entropy, which states that the unstable topological entropy defined in mean u-metrics is the supremum of the unstable metric entropy taken over all invariant measures.

Keywords:

• Unstable topological entropy
• mean u-metrics
• Katok's entropy formula
• variational principle
• partially hyperbolic systems

Acknowledgments

We would like to express our gratitude to Tianyuan Mathematical Center in Southwest China, Sichuan University and Southwest Jiaotong University for their support and hospitality.

Disclosure statement

No potential conflict of interest was reported by the author(s).

Additional information

Funding

The first author was supported by the National Natural Science Foundation (NNSF) of China [11401581, 11971236]. The second author was supported by NNSF of China [11401581]. And the third author was supported by NNSF of China [11671208, 11431012].