Volume growth and entropy for $C^1$ partially hyperbolic diffeomorphisms

Radu Saghin

doi:10.3934/dcds.2014.34.3789

Article Contents

2014, Volume 34, Issue 9: 3789-3801. Doi: 10.3934/dcds.2014.34.3789

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Volume growth and entropy for $C^1$ partially hyperbolic diffeomorphisms

Radu Saghin^1,

1.
Instituto de Matemáticas, Pontifícia Universidad Católica de Valparaíso, Blanco Viel 596, Cerro Barón, Valparaíso

Received: March 31, 2013

Revised: November 30, 2013

Published: August 2014

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Abstract

We show that the metric entropy of a $C^1$ diffeomorphism with a dominated splitting and with the dominating bundle uniformly expanding is bounded from above by the integrated volume growth of the dominating (expanding) bundle plus the maximal Lyapunov exponent from the dominated bundle multiplied by its dimension. We also discuss different types of volume growth that can be associated with an expanding foliation and relationships between them, specially when there exists a closed form non-degenerate on the foliation. Some consequences for partially hyperbolic diffeomorphisms are presented.

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Mathematics Subject Classification: Primary: 37D30, 37A35, 37C40.

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