Smooth conjugacy classes of 3D Axiom A flows

TL;DR

This paper proves a rigidity result for 3D contact Axiom A flows, showing that orbit equivalence with matching periodic orbit lengths implies smooth conjugacy respecting the contact structure, with applications to billiard maps.

Contribution

It extends previous rigidity results to 3D contact Axiom A flows, establishing smooth conjugacy under orbit equivalence and length spectrum conditions.

Findings

Orbit equivalent flows with matching periodic orbit lengths are smoothly conjugate.

The conjugacy respects the contact structure of the flows.

Application to billiard maps shows conjugacy for systems with same length spectrum.

Abstract

We show a rigidity result for 3-dimensional contact Axiom A flows: given two 3D contact Axiom A flows $\Phi_1,\Phi_2$ whose restrictions $\Phi_1|_{\Lambda_1},\Phi_2|_{\Lambda_2}$ to basic sets $\Lambda_1,\Lambda_2$ are orbit equivalent, we prove that if periodic orbits in correspondence have the same length, then the conjugacy is as regular as the flows and respects the contact structure, extending a previous result due to Feldman-Ornstein [21]. Some of the ideas are reminiscent of the work of Otal [51]. As an application, we show that the billiard maps of two open dispersing billiards without eclipse and with the same marked length spectrum are smoothly conjugated.