Dolgopyat type estimates for pinched open billiard flows (Revised)

In this paper we consider open billiard flows in Euclidean spaces with C^1 (un)stable laminations over their non-wandering sets. We show that for such billiard flows the standard symplectic form satisfies a specific non-degeneracy condition over the non-wandering set. This allows to use some previous general results and obtain Dolgopyat type estimates for spectra of Ruelle transfer operators under simpler conditions. We also describe a class of open billiard flows in Euclidean spaces satisfying a certain pinching condition, which in turn implies that the (un)stable laminations over the non-wandering set are C^1.