Mildly dissipative diffeomorphisms of the disk with zero entropy
Dynamical Systems
2023-02-14 v3
Abstract
We discuss the dynamics of smooth diffeomorphisms of the disc with vanishing topological entropy which satisfy the mild dissipation property introduced in [CP]. In particular it contains the H\'enon maps with Jacobian up to 1/4. We prove that these systems are either (generalized) Morse Smale or infinitely renormalizable. In particular we prove for this class of diffeomorphisms a conjecture of Tresser: any diffeomorphism in the interface between the sets of systems with zero and positive entropy admits doubling cascades. This generalizes for these surface dynamics a well known consequence of Sharkovskii's theorem for interval maps.
Cite
@article{arxiv.2005.14278,
title = {Mildly dissipative diffeomorphisms of the disk with zero entropy},
author = {Sylvain Crovisier and Enrique Pujals and Charles Tresser},
journal= {arXiv preprint arXiv:2005.14278},
year = {2023}
}
Comments
This updated version has been accepted for publication in Acta Mathematica