Fixed points of discrete nilpotent group actions on $S^2$

Druck, S.; Fang, Fuquan; Firmo, S. de S.

doi:10.5802/aif.1912

Cited by 15 publications

(31 citation statements)

References 9 publications

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“…On the other hand, these curves can not bound disks in S. Indeed, in that case, using the arguments in [1] or in [3] we prove that f 1 , . .…”

Section: The Main Lemmamentioning

confidence: 86%

“…On the other hand it follows from the topology of S that there exist two distinct curves Γ qi hi and Γ qj hj in the list 6.1.1 which are homotopic. Moreover, they can not bound disks in S since the arguments in [1] or in [3] prove that f 1 , . .…”

mentioning

confidence: 99%

“…, h 2N +n+1 have common fixed points in the interior of that disk, which is a contradiction. Here, we are using the fact that the existence of common fixed points for k commuting diffeomorphisms of Σ restricted to disks, as referred above, does not depend on k. This is a consequence of the arguments developed in [1] or [3].…”

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confidence: 99%

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A note on commuting diffeomorphisms on surfaces

Firmo¹

2005

Nonlinearity

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Abstract. Let Σ be a closed surface with nonzero Euler characteristic. We prove that there exists an open neighborhood V of the identity map of Σ , in the C 1 topology such that: if G is an abelian subgroup of Diff 1 (Σ) generated by any family of elements in V then the elements of G have common fixed points. This result generalizes a similar result of Bonatti announced in his paper Difféomorphismes commutants des surfaces et stabilité des fibrations en tores.

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“…On the other hand, these curves can not bound disks in S. Indeed, in that case, using the arguments in [1] or in [3] we prove that f 1 , . .…”

Section: The Main Lemmamentioning

confidence: 86%

mentioning

confidence: 99%

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A note on commuting diffeomorphisms on surfaces

Firmo¹

2005

Nonlinearity

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“…For the case of groups of circle diffeomorphisms, there is a large variety of results in the same spirit, specially in the real-analytic case [20,67,91,208,209]. One of the main motivations is a beautiful theorem obtained by Duminy at the end of the seventies.…”

Section: A Sharp C 1 Version Via Lyapunov Exponentsmentioning

confidence: 99%

Groups of Circle Diffeomorphisms

Navas¹

2011

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The original version (in Spanish) of this text was prepared for a mini-course in Antofagasta, Chile. Subsequently, enlarged and revised versions were published in the series Monografías del IMCA (Perú) and Ensaios Matemáticos (Brasil). This translation arose from the necessity of making this text accessible to a larger audience. I would like to thank Juan Rivera-Letelier for his invitation to the II Workshop on Dynamical Systems ( 2001), for which the original version was prepared, and Roger Metzger for his invitation to IMCA (2006), where part of this material was presented. I would also like to thank both Étienne Ghys and Maria Eulália Vares for motivating me and allowing me to publish the revised Spanish version in Brasil, as well as all the people who strongly encouraged me to conclude this English version.This work was partially supported by CONICYT and PBCT via the Research Network on Low Dimensional Dynamics. I would also to acknowledge the support of both the UMPA Department of the École Normale Supérieure de Lyon, where the idea of writing this text was born during my PhD thesis, and the Institut des Hautes Études Scientifiques, where the original notes started taking its definite form while I benefited from a one-year postdoctoral position.This work owes a lot to many of my colleges. Several remarks spread throughout the text and the content of some examples and exercises were born during fruitful and quite stimulating discussions. It is then a pleasure to thank Sylvain Crovisier (Proposition 4.2.25),

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“…Our main result concerning S 2 is a generalization to arbitrary abelian subgroups of Diff Druck, Fang and Firmo [5] generalized Bonatti's result to include nilpotent groups. We conjecture that Theorem 1.1 will also generalize to the nilpotent case.…”

Section: Introduction and Notationmentioning

confidence: 99%