On induced systems and ergodicity for holomorphic correspondences

Abstract

The Poincaré recurrence theorem for a holomorphic correspondence provides a natural framework for studying the first return time and a dynamical system thus induced, in this setting. After establishing a version of the Kac's theorem on the average of the first return times for a holomorphic correspondence, we introduce the concept of an induced correspondence, akin to the case of dynamics of maps and prove that it inherits the ergodicity of the holomorphic correspondence. Additionally, we characterise the ergodic measures with respect to the considered holomorphic correspondence, establish the density of forward orbits under mild conditions and prove a Rokhlin-Kakutani type lemma.