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ahlfors regular metric space

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Entropy in uniformly quasiregular dynamics
Part of
ILMARI KANGASNIEMI, YÛSUKE OKUYAMA, PEKKA PANKKA, TUOMAS SAHLSTEN
Journal:

Ergodic Theory and Dynamical Systems / Volume 41 / Issue 8 / August 2021

Published online by Cambridge University Press:

22 June 2020, pp. 2397-2427

Print publication:

August 2021
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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.