Entropy in uniformly quasiregular dynamics

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 \(f\colon M\to M\), the topological entropy \(h(f)\) is \(\log \mathrm{deg}( f )\). This proves Shub's entropy conjecture in this case.