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 $f\colon M\to M$,thetopologicalentropy$h(f)$is$\log $\mathrm{d}\mathrm{e}\mathrm{g}$( f )$. This proves Shub’s entropy conjecture in this case.