Abstract:
We study the structure of groups of circle diffeomorphisms with the property of fixing nonexpandable points. This property generalizes the local expansivity property, and at present there are no known examples of minimal C2 actions of finitely generated groups of circle diffeomorphisms for which this generalized property does not hold.
It turns out that if this property holds for a group action and there is at least one nonexpandable point, then the action admits a rather restrictive characterization. In particular, for such an action, we prove the existence of a Markov partition, and the structure of the action turns out to be similar to that of the Thompson group.
Keywords:
dynamical system, group action, circle diffeomorphism, Markov partition.
Citation:
V. A. Kleptsyn, D. A. Filimonov, “Structure of Groups of Circle Diffeomorphisms with the Property of Fixing Nonexpandable Points”, Funktsional. Anal. i Prilozhen., 46:3 (2012), 38–61; Funct. Anal. Appl., 46:3 (2012), 191–209
\Bibitem{KleFil12}
\by V.~A.~Kleptsyn, D.~A.~Filimonov
\paper Structure of Groups of Circle Diffeomorphisms with the Property of Fixing Nonexpandable Points
\jour Funktsional. Anal. i Prilozhen.
\yr 2012
\vol 46
\issue 3
\pages 38--61
\mathnet{http://mi.mathnet.ru/faa3082}
\crossref{https://doi.org/10.4213/faa3082}
\mathscinet{http://mathscinet.ams.org/mathscinet-getitem?mr=3075039}
\zmath{https://zbmath.org/?q=an:06207361}
\elib{https://elibrary.ru/item.asp?id=20730660}
\transl
\jour Funct. Anal. Appl.
\yr 2012
\vol 46
\issue 3
\pages 191--209
\crossref{https://doi.org/10.1007/s10688-012-0025-1}
\isi{https://gateway.webofknowledge.com/gateway/Gateway.cgi?GWVersion=2&SrcApp=Publons&SrcAuth=Publons_CEL&DestLinkType=FullRecord&DestApp=WOS_CPL&KeyUT=000308818600003}
\elib{https://elibrary.ru/item.asp?id=20482685}
\scopus{https://www.scopus.com/record/display.url?origin=inward&eid=2-s2.0-84866442159}
Linking options:
https://www.mathnet.ru/eng/faa3082
https://doi.org/10.4213/faa3082
https://www.mathnet.ru/eng/faa/v46/i3/p38
This publication is cited in the following 4 articles:
Alvarez S., Filimonov D., Kleptsyn V., Malicet D., Coton C.M., Navas A., Triestino M., “Groups With Infinitely Many Ends Acting Analytically on the Circle”, J. Topol., 12:4 (2019), 1315–1367
Eskif A., Rebelo J.C., “Global Rigidity of Conjugations For Locally Non-Discrete Subgroups of Diff( )(Omega)(S-1)”, J. Mod. Dyn., 15 (2019), 41–93
Deroin B., Kleptsyn V., Navas A., “On the Ergodic Theory of Free Group Actions By Real-Analytic Circle Diffeomorphisms”, Invent. Math., 212:3 (2018), 731–779
D. A. Filimonov, V. A. Kleptsyn, “One-end finitely presented groups acting on the circle”, Nonlinearity, 27:6 (2014), 1205–1223