Real entropy rigidity under quasi-conformal deformations

Khashayar Filom

Published 2018-03-12Version 1

We set up a real entropy function $h_\Bbb{R}$ on the space $\mathcal{M}'_d$ of M\"obius conjugacy classes of real rational maps of degree $d$ by assigning to each class the real entropy of a representative $f\in\Bbb{R}(z)$; namely, the topological entropy of its restriction $f\restriction_{\hat{\Bbb{R}}}$ to the real circle. We prove a structure theorem for the real Julia set $\mathcal{J}_\Bbb{R}(f):=\mathcal{J}(f)\cap\hat{\Bbb{R}}$ that will be utilized to establish a rigidity result stating that $h_\Bbb{R}$ is locally constant on the subspace determined by real maps which are quasi-conformally conjugate to $f$. We also compare the real locus $\mathcal{M}_d(\Bbb{R})$ of the moduli space $\mathcal{M}_d(\Bbb{C})$ of degree $d$ rational maps to the locus $\mathcal{M}'_d$ of classes with a real representative, and we characterize maps of maximal real entropy.