A local ergodic theorem for non-uniformly hyperbolic symplectic maps with singularities (Q2842223)

In this paper, the authors prove a criterion for the local ergodicity of non-uniformly hyperbolic symplectic maps with singularities, which model billiards and other physical systems. Their main result gives sufficient conditions for a point in the domain of such an invertible map \(F\) (where \(F\) preserves a symplectic form and is non-uniformly hyperbolic), to have a neighbourhood contained (up to a set of measure zero), in one ergodic component of \(F\). Local ergodic results can be then employed to prove ergodicity of non-uniformly hyperbolic systems. From their local ergodic theorem, the authors derive another criterion for the ergodicity of \(F\), based on the topology of the set \(X\) of the points to which the above local ergodic theorem applies. Their result extends a theorem of \textit{C. Liverani} and \textit{M. P. Wojtkowski} [in: Dynamics reported. Expositions in dynamical systems. New series. Vol. 4. Berlin: Springer-Verlag. 130--202 (1995; Zbl 0824.58033)].