Diffeotopically trivial periodic diffeomorphisms (original) (raw)
1970, Inventiones Mathematicae
https://doi.org/10.1007/BF01403188
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Abstract
In this paper, we settle negatively an old question as to whether all free periodic diffeomorphisms that are diffeotopic to the identity can be found by restricting circle group actions to finite cyclic subgroups. More precisely, we construct examples of periodic diffeomorphisms of D 2" for n>3 which on S z"-~ are free and cannot be obtained from any piecewise linear (PL) circle group action on S 2"-1 (cf. Gluck [5]).
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Following the recent advances in the study of groups of circle diffeomorphisms, we classify the topological dynamics of locally discrete, finitely generated, virtually free subgroups of the group Diff ω + (S 1 ) of orientation preserving real-analytic circle diffeomorphisms, which include all subgroups of Diff ω + (S 1 ) acting with an invariant Cantor set. An important tool that we develop, of independent interest, is the extension of classical ping-pong lemma to actions of fundamental groups of graphs of groups. Our main motivation is an old conjecture by P.R. Dippolito [Ann. Math. 107 (1978), 403-453] from foliation theory, which we solve in this restricted but significant setting: this and other consequences of the classification will be treated in more detail in a companion work.
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Periodic maps on R7 without fixed points
Mathematical Proceedings of The Cambridge Philosophical Society, 2002
In this paper we prove that R 7 admits smooth periodic maps with no fixed points for every period that is not a prime power. Results of P. A. Smith show that such examples do not exist in any lower dimensions.
arXiv (Cornell University), 2021
In the first part of this work we have established an efficient method to obtain a topological classification of locally discrete, finitely generated, virtually free subgroups of real-analytic circle diffeomorphisms. In this second part we describe several consequences, among which the solution (within this setting) to an old conjecture by P. R. Dippolito [Ann. Math. 107 (1978), 403-453] that actions with invariant Cantor sets must be semi-conjugate to piecewise linear actions. In addition, we exhibit examples of locally discrete, minimal actions which are not of Fuchsian type.
1989
The periodic orbit structure of orientation preserving diffeomorphisms on D2 with topological entropy zero Annales de l'I. H. P., section A, tome 50, n o 3 (1989), p. 335-356 http://www.numdam.org/item?id=AIHPA\_1989\_\_50\_3\_335\_0 © Gauthier-Villars, 1989, tous droits réservés. L'accès aux archives de la revue « Annales de l'I. H. P., section A » implique l'accord avec les conditions générales d'utilisation (http://www.numdam. org/conditions). Toute utilisation commerciale ou impression systématique est constitutive d'une infraction pénale. Toute copie ou impression de ce fichier doit contenir la présente mention de copyright. Article numérisé dans le cadre du programme Numérisation de documents anciens mathématiques
On a factorization of homeomorphisms of the circle possessing periodic points
Journal of Mathematical Analysis and Applications, 2008
We prove that for every orientation-preserving homeomorphism F : S 1 → S 1 possessing periodic points of order n there exist a homeomorphism T : S 1 → S 1 such that T n = id and a homeomorphism G : S 1 → S 1 without periodic points except fixed points such that
Topological conjugacy of circle diffeomorphisms
Ergodic Theory and Dynamical Systems, 1997
The classical criterion for a circle diffeomorphism to be topologically conjugate to an irrational rigid rotation was given by A. Denjoy [1]. In [5] one of us gave a new criterion. There is an example satisfying Denjoy's bounded variation condition rather than [5]'s Zygmund condition and vice versa. This paper will give the third criterion which is implied by either of the above criteria.
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References (11)
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- Charles H. Giffen Mathematisches Institut der Universit/it BRD-6900 Heidelberg 1 TiergartenstraBe Germany (Received June 22, 1970)
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