A link between Topological Entropy and Lyapunov Exponents

TL;DR

This paper establishes a fundamental link between topological entropy and Lyapunov exponents for generic symplectic diffeomorphisms, showing entropy can be characterized by periodic points and approximated by hyperbolic sets.

Contribution

It proves that for generic non-partially hyperbolic symplectic diffeomorphisms, topological entropy equals the supremum of positive Lyapunov exponents at hyperbolic periodic points and is lower semicontinuous.

Findings

Topological entropy equals the supremum of positive Lyapunov exponents for generic systems.

Entropy can be approximated by entropy on basic hyperbolic sets.

Entropy map is lower semicontinuous in a generic set of symplectic diffeomorphisms.

Abstract

We show that a $C^1-$generic non partially hyperbolic symplectic diffeomorphism $f$ has topological entropy equal to the supremum of the sum of the positive Lyapunov exponents of its hyperbolic periodic points. Moreover, we also prove that $f$ has topological entropy approximated by the topological entropy of $f$ restrict to basic hyperbolic sets. In particular, the topological entropy map is lower semicontinuous in a $C^1-$generic set of symplectic diffeomorphisms far from partial hyperbolicity.