A new criterion of physical measures for partially hyperbolic diffeomorphisms
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- by Yongxia Hua, Fan Yang and Jiagang Yang PDF
- Trans. Amer. Math. Soc. 373 (2020), 385-417 Request permission
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
- MR Author ID: 841820
- 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
- MR Author ID: 949931
- Email: yangjg@impa.br
- 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 - © Copyright 2019 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 373 (2020), 385-417
- MSC (2010): Primary 37D30, 37C40, 37A35
- DOI: https://doi.org/10.1090/tran/7920
- MathSciNet review: 4042879