$C1$-Genericity of Symplectic Diffeomorphisms and Lower Bounds for Topological Entropy

TL;DR

This paper demonstrates that for a large class of symplectic diffeomorphisms, the topological entropy is bounded below by Lyapunov exponents of certain periodic points, revealing a generic relationship between entropy and hyperbolic dynamics.

Contribution

It establishes a $C^1$-generic lower bound for topological entropy in terms of Lyapunov exponents on a residual set of symplectic diffeomorphisms, extending understanding of entropy in symplectic dynamics.

Findings

Residual set of symplectic diffeomorphisms with entropy bounds

Trichotomy: Anosov, partially hyperbolic, or dense elliptic points

Existence of elliptic periodic points converging to the manifold

Abstract

There is a $C^1$-residual (Baire second class) subset $\mathcal{R}$ of symplectic diffeomorphisms on $2d$-dimensional manifold, $d\geq 1$, such that for every non-Anosov $f$ in $\mathcal{R}$ its topological entropy is lower bounded by the supremum of the Lyapunov exponents of their hyperbolic periodic points in the \emph{unbreakable central subbundle} (i.e., central direction with no dominated splitting) of $f$. The previous result deals with the fact that for $f$ in a residual set $\tilde{\mathcal{R}}$ of symplectic diffeomorphisms (containing $\mathcal{R}$) satisfies a trichotomy: or $f$ is Anosov or $f$ is robustly transitive partially hyperbolic with {\em unbreakable center} of dimension $2m$, $0 < m < d$, or $f$ has totally elliptic periodic points dense on $M$. In the second case, we also show the existence of a sequence of $m$-{\em elliptic} periodic points converging to $M$. Indeed, $\tilde{\mathcal{R}}$ contains an open and dense subset.