Real entropy rigidity under quasi-conformal deformations

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