Area-preserving diffeomorphisms on the disk and positive hyperbolic orbits

TL;DR

This paper proves that certain area-preserving disk diffeomorphisms with multiple interior periodic points have infinitely many positive hyperbolic periodic points, with applications to contact 3-manifolds and orbit classifications.

Contribution

It establishes a new link between interior periodic points and hyperbolic orbits for non-degenerate area-preserving disk diffeomorphisms and applies this to contact geometry.

Findings

Infinitely many positive hyperbolic periodic points under given conditions

Refinement of orbit classification in non-degenerate contact 3-spheres

Existence of infinitely many simple positive hyperbolic orbits or exactly two elliptic orbits

Abstract

In this paper, we prove that if an area-preserving non-degenerate diffeomorphism on the open disk which extend smoothly to the boundary with non-degeneracy has at least 2 interior periodic points, then there are infinitely many positive hyperbolic periodic points on the interior. As an application, we prove that if a non-degenerate universally tight contact 3-dimentional lens space has a Birkhoff section of disk type and at least 3 simple periodic orbits, there are infinitely many simple positive hyperbolic orbits. In particular, we have that a non-degenerate dynamically convex contact 3-sphere has either infinitely many simple positive hyperbolic orbits or exactly two simple elliptic orbits, which gives a refinement of the result proved by Hofer, Wysocki and Zehnder in \cite{HWZ2} under non-degeneracy.