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Transactions of the American Mathematical Society

Published by the American Mathematical Society since 1900, Transactions of the American Mathematical Society is devoted to longer research articles in all areas of pure and applied mathematics.

ISSN 1088-6850 (online) ISSN 0002-9947 (print)

The 2020 MCQ for Transactions of the American Mathematical Society is 1.48 .

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Robust minimality of strong foliations for DA diffeomorphisms: $cu$-volume expansion and new examples
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by Jana Rodriguez Hertz, Raúl Ures and Jiagang Yang PDF
Trans. Amer. Math. Soc. 375 (2022), 4333-4367 Request permission

Abstract:

Let $f$ be a $C^2$ partially hyperbolic diffeomorphisms of $\mathbb {T}^3$ (not necessarily volume preserving or transitive) isotopic to a linear Anosov diffeomorphism $A$ with eigenvalues \begin{equation*} \lambda _{s}<1<\lambda _{c}<\lambda _{u}. \end{equation*} Under the assumption that the set \begin{equation*} \{x: \,\mid \log \det (Tf\mid _{E^{cu}(x)})\mid \leq \log \lambda _{u} \} \end{equation*} has zero volume inside any unstable leaf of $f$ where $E^{cu} = E^c\oplus E^u$ is the center unstable bundle, we prove that the stable foliation of $f$ is $C^1$ robustly minimal, i.e., the stable foliation of any diffeomorphism $C^1$ sufficiently close to $f$ is minimal. In particular, $f$ is robustly transitive.

We build, with this criterion, a new example of a $C^1$ open set of partially hyperbolic diffeomorphisms, for which the strong stable foliation and the strong unstable foliation are both minimal.

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Additional Information
  • Jana Rodriguez Hertz
  • Affiliation: Department of Mathematics, Southern University of Science and Technology, Shenzhen, Guangdong, People’s Republic of China; and SUSTech International Center for Mathematics, Shenzhen, Guangdong, People’s Republic of China
  • MR Author ID: 657286
  • ORCID: 0000-0002-7683-1869
  • Email: rhertz@sustech.edu.cn
  • Raúl Ures
  • Affiliation: Department of Mathematics, Southern University of Science and Technology, Shenzhen, Guangdong, People’s Republic of China; and SUSTech International Center for Mathematics, Shenzhen, Guangdong, People’s Republic of China
  • ORCID: 0000-0002-3262-8922
  • Email: ures@sustech.edu.cn
  • Jiagang Yang
  • Affiliation: Departamento de Geometria, Instituto de Matemática e Estatística, Universidade Federal Fluminense, Niterói, Brazil
  • MR Author ID: 949931
  • Email: yangjg@impa.br
  • Received by editor(s): December 29, 2019
  • Received by editor(s) in revised form: October 13, 2020, May 19, 2021, October 29, 2021, and November 2, 2021
  • Published electronically: February 3, 2022
  • Additional Notes: The first and second authors were partially supported by NNSFC 11871262. The first author was partially supported by NNSFC 11871394. The second and third authors were partially supported by NNSFC 12071202. The third author was partially supported by CNPq, FAPERJ, and PRONEX of Brazil and NNSFC 11871487 of China. Most of the research for this paper was made during a visit by the third author to SUSTech’s Mathematics Department. The third author is very grateful for the good working environment during his visit and for the support received from the Department colleagues and authorities, in particular from the first and second authors
  • © Copyright 2022 American Mathematical Society
  • Journal: Trans. Amer. Math. Soc. 375 (2022), 4333-4367
  • MSC (2020): Primary 37D30; Secondary 37B20
  • DOI: https://doi.org/10.1090/tran/8590
  • MathSciNet review: 4419061