Symbolic dynamics for normally hyperbolic foliations

ABSTRACT

We introduce some tools of symbolic dynamics to study the hyperbolic directions of partially hyperbolic diffeomorphisms, emulating the well-known methods available for uniformly hyperbolic systems.

Keywords:

• Partially hyperbolic diffeomorphisms
• Markov partitions

MATHEMATICAL SUBJECT CLASSIFICATION CODES:

• 37D30
• 37C85

Acknowledgments

The results presented in this article are generalizations of the ones obtained in my MSc project under the supervision of Mike Shub. I am very grateful to him for all the ideas and encouragement that he gave me. I have also benefited a lot from conversations wit Charles Pugh, Enrique Pujals and Federico Rodriguez-Hertz. The author is particularly indebted to the referee for her/his diligent work, that allowed to remove errors and greatly improved the correctness and readability of the article.

To all of them, my most sincere thanks.

Disclosure statement

No potential conflict of interest was reported by the author.

Notes

1. In the literature, the notion of pseudo-group is usually defined using open sets instead of proper ones, but the extension to the later case is straightforward. Compare [10].

2. A foliated atlas {U} _k is nice if whenever there exists a distinguished open set containing .

Additional information

Funding

The author was supported by FAPESP [project number 2014/119262-0].