The central limit theorem and ergodicity

In this work, some relationships between stochastic properties and topological properties of dynamical systems are investigated. Let ff be a continuous map from a compact metric space XX to itself. We prove that if ff satisfies the central limit theorem, then ff is topologically strongly ergodic and (X,f)(X,f) is an E-system, that is, ff is topologically transitive and there is an invariant Borel probability measure mm with full support.