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Entropy in uniformly quasiregular dynamics

Part of: Entire and meromorphic functions, and related topics Low-dimensional topology Geometric function theory

Published online by Cambridge University Press:  22 June 2020

ILMARI KANGASNIEMI Open the ORCID record for ILMARI KANGASNIEMI [Opens in a new window] ,
YÛSUKE OKUYAMA ,
PEKKA PANKKA Open the ORCID record for PEKKA PANKKA [Opens in a new window]  and
TUOMAS SAHLSTEN Open the ORCID record for TUOMAS SAHLSTEN [Opens in a new window]
ILMARI KANGASNIEMI
Affiliation:
Department of Mathematics and Statistics, P.O. Box 68 (Pietari Kalmin katu 5), FI-00014 University of Helsinki, Finland (e-mail: ilmari.kangasniemi@helsinki.fi, pekka.pankka@helsinki.fi)
YÛSUKE OKUYAMA
Affiliation:
Division of Mathematics, Kyoto Institute of Technology, Sakyo-ku, Kyoto 606-8585, Japan (e-mail: okuyama@kit.ac.jp)
PEKKA PANKKA
Affiliation:
Department of Mathematics and Statistics, P.O. Box 68 (Pietari Kalmin katu 5), FI-00014 University of Helsinki, Finland (e-mail: ilmari.kangasniemi@helsinki.fi, pekka.pankka@helsinki.fi)
TUOMAS SAHLSTEN
Affiliation:
School of Mathematics, University of Manchester, UK (e-mail: tuomas.sahlsten@manchester.ac.uk)
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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.

Keywords

uniformly quasiregular mappingsentropyAhlfors regular metric space

MSC classification

Primary: 30C65: Quasiconformal mappings in $^n$, other generalizations
Secondary: 57M12: Special coverings, e.g. branched 30D05: Functional equations in the complex domain, iteration and composition of analytic functions
Type
Original Article
Information
Ergodic Theory and Dynamical Systems , Volume 41 , Issue 8 , August 2021 , pp. 2397 - 2427
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Copyright
© The Author(s) 2020. Published by Cambridge University Press

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