Smooth conjugacy classes of 3D Axiom A flows

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.

arXiv commentThere was a mistake in Proposition 3.1; it affects the result about spectral rigidity of open dispersing billiards, which was removed from the paper. In the present version, we focus on dynamical results. The main dynamical result has been improved (upgraded regularity). We also discuss the preservation of symmetries by the conjugacy

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