Entropy in uniformly quasiregular dynamics

Abstract

AbstractLet $M$ 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.