Acta Mathematica

Volume 232 (2024)

Number 2

Mildly dissipative diffeomorphisms of the disk with zero entropy

Pages: 221 – 323

Authors

Sylvain Crovisier (Laboratoire de Math\’ematiques, Universit\’e Paris-Saclay, Orsay, France)

Enrique Pujals (Department of Mathematics, City University of New York Graduate Center, New York, NY, U.S.A.)

Charles Tresser (Instituto de Matem\’atica Pura e Aplicada (IMPA), Rio de Janeiro, Brazil)

Abstract

We discuss the dynamics of smooth diffeomorphisms of the disc with vanishing topological entropy which satisfy the mild dissipation property introduced in [CP]. This class contains the H\'enon maps with Jacobian up to $\tfrac14$. We prove that these systems are either (generalized) Morse--Smale or infinitely renormalizable. In particular, we prove a conjecture of Tresser in this class: any diffeomorphism in the interface between the sets of systems with zero and positive entropy admits doubling cascades. This generalizes a well-known consequenceof Sharkovsky's theorem for interval maps to mild dissipative diffeomorphisms of the disk with zero entropy.

The full text of this article has not yet been published online.

S. C. was partially supported by the ERC project 692925 – NUHGD; S. C. and E.P. were partiallysupported by the Balzan Research Project of J. Palis. E.P. received support of NSF via grant DMS-1956022.

Received 23 June 2020

Received revised 10 May 2022

Accepted 18 December 2022

Published 8 October 2024