Entropy in uniformly quasiregular dynamics

Abstract

Let $M$ be a closed, oriented, and connected Riemannian $n$ -manifold, for $n\geq 2$ , which is not a rational homology sphere. We show that, for a non-constant and non-injective uniformly quasiregular self-map $f:M\rightarrow M$ , the topological entropy $h(f)$ is $\log \deg f$ . This proves Shub’s entropy conjecture in this case.

Asymptotically holomorphic methods for infinitely renormalizable unimodal maps

“…This means that our maps are not uniformly quasiconformal, that is, there exists no K so that all iterates are K -quasiconformal. So we cannot apply the rich theory developed for such maps, see for example [8, 35].…”