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

TL;DR

This paper establishes a lower bound for the topological entropy of generic non-Anosov symplectic diffeomorphisms, linking entropy to Lyapunov exponents and exploring properties like symbolic extensions.

Contribution

It proves that for generic symplectic diffeomorphisms, entropy is bounded below by Lyapunov exponents unless the system is Anosov, and shows non-existence of symbolic extensions outside Anosov systems.

Findings

Generic symplectic diffeomorphisms are either Anosov or have entropy bounded by Lyapunov exponents.

Non-Anosov generic symplectic diffeomorphisms lack symbolic extensions.

Examples of volume-preserving diffeomorphisms with discontinuous entropy in $C^1$ topology.

Abstract

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