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Accessibility and centralizers for partially hyperbolic flows

Part of: Dynamical systems with hyperbolic behavior Smooth dynamical systems: general theory

Published online by Cambridge University Press:  11 May 2021

TODD FISHER  and
BORIS HASSELBLATT
TODD FISHER*
Affiliation:
Department of Mathematics, Brigham Young University, Provo, UT 84602, USA
BORIS HASSELBLATT
Affiliation:
Department of Mathematics, Tufts University, Medford, MA 02144, USA (e-mail: boris.hasselblatt@tufts.edu)
*
e-mail: tfisher@math.byu.edu
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Abstract

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Stable accessibility of partially hyperbolic systems is central to their stable ergodicity, and we establish its $C^1$-density among partially hyperbolic flows, as well as in the categories of volume-preserving, symplectic, and contact partially hyperbolic flows. As applications, we obtain on one hand in each of these four categories of flows the $C^1$-density of the $C^1$-stable topological transitivity and triviality of the centralizer, and on the other hand the $C^1$-density of the $C^1$-stable K-property of the natural volume in the latter three categories.

Keywords

accessibilitycentralizerdynamical systemspartially hyperbolic systems and dominated splittings

MSC classification

Primary: 37C80: Symmetries, equivariant dynamical systems
Secondary: 37D30: Partially hyperbolic systems and dominated splittings 37D40: Dynamical systems of geometric origin and hyperbolicity (geodesic and horocycle flows, etc.)

Information

Type
Original Article
Information
Ergodic Theory and Dynamical Systems , Volume 42 , Issue 3: Anatole Katok Memorial Issue Part 2: Special Issue of Ergodic Theory and Dynamical Systems , March 2022 , pp. 835 - 854
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Copyright
© The Author(s), 2021. Published by Cambridge University Press

Footnotes

To the memory of Anatole Katok

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