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Transactions of the American Mathematical Society

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A new criterion of physical measures for partially hyperbolic diffeomorphisms


Authors: Yongxia Hua, Fan Yang and Jiagang Yang
Journal: Trans. Amer. Math. Soc. 373 (2020), 385-417
MSC (2010): Primary 37D30, 37C40, 37A35
DOI: https://doi.org/10.1090/tran/7920
Published electronically: August 20, 2019
MathSciNet review: 4042879
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Abstract: We show that, for any $ C^1$ partially hyperbolic diffeomorphism, there is a full volume subset such that any Cesàro limit of any point in this subset satisfies the Pesin formula for partial entropy.

This result has several important applications. First, we show that, for any $ C^{1+}$ partially hyperbolic diffeomorphism with one dimensional center, there is a full volume subset such that either every point in this set belongs to the basin of a physical measure with nonvanishing center exponent or the center exponent of any limit of the sequence $ \frac 1n\sum _{i=0}^{n-1}\delta _{f^i(x)}$ is vanishing.

We also prove that, for any diffeomorphism with mostly contracting center, it admits a $ C^1$-neighborhood such that every diffeomorphism in a $ C^1$ residual subset of this open set admits finitely many physical measures whose basins have full volume.


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Additional Information

Yongxia Hua
Affiliation: Department of Mathematics, Southern University of Science and Technology of China, 1088 Xueyuan Road, Xili, Nanshan District, Shenzhen, Guangdong 518055, People’s Republic of China
Email: huayx@sustc.edu.cn

Fan Yang
Affiliation: Department of Mathematics, University of Oklahoma, Norman, Oklahoma 73019
Email: fan.yang-2@ou.edu

Jiagang Yang
Affiliation: Departamento de Geometria, Instituto de Matemática e Estatística, Universidade Federal Fluminense, Niterói, Brazil
Address at time of publication: Department of Mathematics, Southern University of Science and Technology of China, Guangdong, People’s Republic of China
Email: yangjg@impa.br

DOI: https://doi.org/10.1090/tran/7920
Received by editor(s): February 27, 2019
Received by editor(s) in revised form: May 28, 2019
Published electronically: August 20, 2019
Additional Notes: The first author was supported by NSFC Grant No. 11401133.
The third author was partially supported by CNPq, FAPERJ, and PRONEX
Article copyright: © Copyright 2019 American Mathematical Society