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Entropy in uniformly quasiregular dynamics

Published online by Cambridge University Press:  22 June 2020

ILMARI KANGASNIEMI
Affiliation:
Department of Mathematics and Statistics, P.O. Box 68 (Pietari Kalmin katu 5), FI-00014 University of Helsinki, Finland (e-mail: [email protected], [email protected])
YÛSUKE OKUYAMA
Affiliation:
Division of Mathematics, Kyoto Institute of Technology, Sakyo-ku, Kyoto 606-8585, Japan (e-mail: [email protected])
PEKKA PANKKA
Affiliation:
Department of Mathematics and Statistics, P.O. Box 68 (Pietari Kalmin katu 5), FI-00014 University of Helsinki, Finland (e-mail: [email protected], [email protected])
TUOMAS SAHLSTEN
Affiliation:
School of Mathematics, University of Manchester, UK (e-mail: [email protected])

Abstract

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.

Type
Original Article
Copyright
© The Author(s) 2020. Published by Cambridge University Press

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