Robustness of topological entropy under small area deformations

Abstract

In this paper, we establish a new type of stability phenomenon for the topological entropy of Hamiltonian diffeomorphisms of closed surfaces. For a closed surface endowed with an area form $(\Sigma ,\omega )$ and a Hamiltonian diffeomorphism $\varphi$ of $(\Sigma ,\omega )$, we show that for every $\epsilon >0$ there exists $A=A(\varphi ,\epsilon )>0$ such that \[ h_{\mathrm{top}}(ϕ') > h_{\mathrm{top}}(ϕ)-\varepsilon \] for every Hamiltonian diffeomorphism ${\varphi }^{\prime }$ obtained from $\varphi$ by a deformation supported in a disjoint union of disks, each of area less than $A$. In particular, if $h_{\mathrm{top}}(\varphi )>0$, then $\varphi$ cannot be made to have zero entropy by an area-preserving deformation supported in disks of small area. This follows from the new braid stability result established in this paper with respect to the spectral distance recently introduced by Connery-Grigg.

92 pages, 9 figures