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Proper extensions of the 2-sphere’s conformal group present entropy and are 4-transitive

Part of: Low-dimensional topology Topological dynamics

Published online by Cambridge University Press:  03 May 2023

ULISSES LAKATOS Open the ORCID record for ULISSES LAKATOS [Opens in a new window]  and
FÁBIO TAL Open the ORCID record for FÁBIO TAL [Opens in a new window]
ULISSES LAKATOS*
Affiliation:
Instituto de Matemática e Estatística, Universidade de São Paulo, Rua do Matão 1010, Cidade Universitária, 05508-090 São Paulo, SP, Brazil (e-mail: fabiotal@ime.usp.br)
FÁBIO TAL
Affiliation:
Instituto de Matemática e Estatística, Universidade de São Paulo, Rua do Matão 1010, Cidade Universitária, 05508-090 São Paulo, SP, Brazil (e-mail: fabiotal@ime.usp.br)
*
e-mail: lakatos@ime.usp.br
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Abstract

In this paper, we prove using elementary techniques that any group of diffeomorphisms acting on the 2-sphere and properly extending the conformal group of Möbius transformations must be at least 4-transitive or, more precisely, arc 4-transitive. As an important consequence, we derive that any such group must always contain an element of positive topological entropy. We also provide a self-contained characterization, in terms of transitivity, of the Möbius transformations within the full group of sphere diffeomorphisms.

Keywords

transitivitygroup actions on surfacestopological groupstopological entropy

MSC classification

Primary: 57M60: Group actions in low dimensions
Secondary: 37B05: Transformations and group actions with special properties (minimality, distality, proximality, etc.)
Type
Original Article
Information
Ergodic Theory and Dynamical Systems , Volume 44 , Issue 4 , April 2024 , pp. 1102 - 1122
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Copyright
© The Author(s), 2023. Published by Cambridge University Press

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