Ergodic Theory and Dynamical Systems

We study the family of one dimensional piecewise smooth dynamical systems in which two classic theorems are still permanent. One of them is Birkhoff Transitivity Theorem and the other one is Banks, Brooks, Cairns, Davis and Stacey Theorem. Baker like maps with N-branches ( N>2 ) constitute an open subset of this family. We show that under certain conditions, a full chaos always occurs in the family of Baker like maps.