Infimum of the metric entropy of volume preserving Anosov systems

Huyi Hu; Miaohua Jiang; Yunping Jiang

doi:10.3934/dcds.2017205

Article Contents

2017, Volume 37, Issue 9: 4767-4783. Doi: 10.3934/dcds.2017205

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Infimum of the metric entropy of volume preserving Anosov systems

1.
Department of Mathematics, Michigan State University, East Lansing, MI 48824, USA
2.
Department of Mathematics and Statistics, Wake Forest University, Winston Salem, NC 27109, USA
3.
Department of Mathematics, Queens College of the City University of New York, Flushing, NY 11367, USA
4.
Department of Mathematics, Graduate Center of the City University of New York, New York, NY 10016, USA

* Corresponding author: Miaohua Jiang
* Corresponding author: Miaohua Jiang

Received: April 30, 2016

Revised: March 31, 2017

Published: October 2017

Y. Jiang is partially supported by the collaboration grant from the Simons Foundation [grant number 199837] and awards from PSC-CUNY and grants from NSFC [grant numbers 11171121 and 11571122]

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Abstract

In this paper we continue our study [9] of the infimum of the metric entropy of the SRB measure in the space of hyperbolic dynamical systems on a smooth Riemannian manifold of higher dimension. We restrict our study to the space of volume preserving Anosov diffeomorphisms and the space of volume preserving expanding endomorphisms. In our previous paper, we use the perturbation method at a hyperbolic periodic point. It raises the question whether the volume can be preserved. In this paper, we answer this question affirmatively. We first construct a smooth path starting from any point in the space of volume preserving Anosov diffeomorphisms such that the metric entropy tends to zero as the path approaches the boundary of the space. Similarly, we construct a smooth path starting from any point in the space of volume preserving expanding endomorphisms with a fixed degree greater than one such that the metric entropy tends to zero as the path approaches the boundary of the space. Therefore, the infimum of the metric entropy as a functional is zero in both spaces.

Keywords:

Volume preserving Anosov diffeomorphism,
volume preserving expanding endomorphism,
metric entropy,
infimum of metric entropy.

Mathematics Subject Classification: Primary: 37C40; Secondary: 37A35, 37D20.

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References

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