Real entropy rigidity under quasi-conformal deformations

Abstract

We set up a real entropy function $h_{\mathbb{R}}$ on the space ${\mathcal{M}}_d^{\prime }$ of Möbius conjugacy classes of real rational maps of degree $d$ by assigning to each class the real entropy of a representative $f\in \mathbb{R}(z)$; namely, the topological entropy of its restriction $f{\upharpoonright }_{\hat{\mathbb{R}}}$ to the real circle. We prove a rigidity result stating that $h_{\mathbb{R}}$ is locally constant on the subspace determined by real maps quasi-conformally conjugate to $f$. As examples of this result, we analyze real analytic stable families of hyperbolic and flexible Lattès maps with real coefficients along with numerous families of degree $d$ real maps of real entropy $\log (d)$. The latter discussion moreover entails a complete classification of maps of maximal real entropy.

Minor revisions. New references [Bro76], [HQ15] and [Mis81] are added. To appear in Conformal Geometry and Dynamics. The published version is shorter and does not involve the appendices. 59 pages and 3 figures