Non-integrability of open billiard flows and Dolgopyat-type estimates

Abstract

Abstract We consider open billiard flows in ℝ n and show that the standard symplectic form dα in ℝ n satisfies a specific non-integrability condition over their non-wandering sets Λ. This allows one to use the main result in Stoyanov [Spectra of Ruelle transfer operators for Axiom A flows. Preprint , 2010, arXiv:0810.1126v4 [math.DS]] and obtain Dolgopyat-type estimates for the spectra of Ruelle transfer operators under simpler conditions. We also describe a class of open billiard flows in ℝ n ( n ≥3) satisfying a certain pinching condition, which in turn implies that the (un)stable laminations over the non-wandering set are C 1 .