Recurrence in the dynamics of meromorphic correspondences and holomorphic semigroups

Ergodicity in the dynamics of holomorphic correspondences

“…Remark 4.2. Given a meromorphic correspondences F on a complex manifold X and a Borel subset B of X, F † (B) need not be a Borel subset of X (see [15] for examples). This difficulty can be handled using tools from descriptive set theory.…”

“…This difficulty can be handled using tools from descriptive set theory. Using these tools further as in [15], Theorem 1.2 can be proved when F is defined on a compact manifold and for an F * invariant measure that puts zero mass on pluripolar sets. The above proof of Theorem 1.5 also holds with appropriate changes for the meromorphic case.…”