Robustness of topological entropy under small area deformations

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 $(Σ,ω)$ and a Hamiltonian diffeomorphism $φ$ of $(Σ,ω)$, we show that for every $\varepsilon>0$ there exists $A=A(φ,\varepsilon)>0$ such that \[ h_{\mathrm{top}}(φ') > h_{\mathrm{top}}(φ)-\varepsilon \] for every Hamiltonian diffeomorphism $φ'$ obtained from $φ$ by a deformation supported in a disjoint union of disks, each of area less than $A$. In particular, if $h_{\mathrm{top}}(φ)>0$, then $φ$ 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.