A lower bound for topological entropy of generic non-Anosov symplectic diffeomorphisms

Abstract

We prove that a 'C POT.1' generic symplectic diffeomorphism is either Anosov or its topological entropy is bounded from below by the supremum over the smallest positive Lyapunov exponent of its periodic points. We also prove that 'C POT.1' generic symplectic diffeomorphisms outside the Anosov ones do not admit symbolic extension and, finally, we give examples of volume preserving surface diffeomorphisms which are not points of upper semicontinuity of the entropy function in the 'C POT.1' topology.