Entropy in uniformly quasiregular dynamics

Abstract

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

31 pages, v3: streamlined the proof of the entropy lower bound based on comments from P. Haïssinsky. Also stated some more general versions of the lower bound result, and fixed various typos throughout the paper