Mildly dissipative diffeomorphisms of the disk with zero entropy

TL;DR

This paper studies smooth disk diffeomorphisms with zero entropy and mild dissipation, showing they are either Morse-Smale or infinitely renormalizable, and confirms a conjecture relating zero and positive entropy systems.

Contribution

It proves a conjecture of Tresser for surface dynamics, linking zero entropy systems with doubling cascades, extending Sharkovskii's theorem.

Findings

Diffeomorphisms are either Morse-Smale or infinitely renormalizable.

Systems at the interface between zero and positive entropy admit doubling cascades.

Includes Hénon maps with Jacobian up to 1/4 as examples.

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.