Normal forms in a neighborhood of hyperbolic periodic orbits for flows in dimension 3

TL;DR

This paper develops normal form theories for volume-preserving and contact flows near hyperbolic periodic orbits in 3D, leading to new rigidity results and insights into the local dynamics of such systems.

Contribution

It introduces novel normal form constructions for generators of flows and contact forms near hyperbolic orbits, and establishes new local rigidity results for 3D contact flows.

Findings

Existence of normal forms for flow generators near hyperbolic periodic orbits.

Normal forms for contact forms and Reeb vector fields in the neighborhood of hyperbolic orbits.

A new local rigidity result linking roof functions and return maps in 3D contact flows.

Abstract

In a neighborhood of a hyperbolic periodic orbit of a volume-preserving flow on a manifold of dimension 3, we define and show the existence of a normal form for the generator of the flow that encodes the dynamics. If the flow is a contact flow, we show the existence of a normal form for the contact form what results in an improved normal form for its Reeb vector field. Additionally, we present a few rigidity results associated to periodic data for Anosov contact flows derived from the underlying normal form theory. Finally, we establish a new local rigidity result for contact flows on manifolds of dimension 3 in a neighborhood of a hyperbolic periodic point by finding a new link between the roof function and the return map to a section.