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Simultaneous dense and non-dense orbits for certain partially hyperbolic diffeomorphisms

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

Published online by Cambridge University Press:  04 September 2018

WEISHENG WU
WEISHENG WU*
Affiliation:
Department of Applied Mathematics, College of Science, China Agricultural University, Beijing, 100083, PR China email wuweisheng@cau.edu.cn
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Abstract

Let $g:M\rightarrow M$ be a $C^{1+\unicode[STIX]{x1D6FC}}$-partially hyperbolic diffeomorphism preserving an ergodic normalized volume on $M$. We show that, if $f:M\rightarrow M$ is a $C^{1+\unicode[STIX]{x1D6FC}}$-Anosov diffeomorphism such that the stable subspaces of $f$ and $g$ span the whole tangent space at some point on $M$, the set of points that equidistribute under $g$ but have non-dense orbits under $f$ has full Hausdorff dimension. The same result is also obtained when $M$ is the torus and $f$ is a toral endomorphism whose center-stable subspace does not contain the stable subspace of $g$ at some point.

Keywords

Nondense orbitsSchmidt gamespartial hyperbolicitydimension theory

MSC classification

Primary: 37D30: Partially hyperbolic systems and dominated splittings
Secondary: 37C45: Dimension theory of dynamical systems
Type
Original Article
Information
Ergodic Theory and Dynamical Systems , Volume 40 , Issue 4 , April 2020 , pp. 1083 - 1107
Copyright
© Cambridge University Press, 2018

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