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We provide a topological proof that each orientation reversing homeomorphism of the 2-sphere which has a point of period k ≥ 3 also has a point of period 2. Moreover if such a k-periodic point can be chosen arbitrarily close to an isolated fixed point o then the same is true for the 2-periodic point. We also strengthen this result by proving that if an orientation reversing homeomorphism h of the sphere has no 2-periodic point then the complement of the fixed point set can be covered by invariant...
We prove that the standard action of the mapping class group of a surface of sufficiently large genus on the unit tangent bundle is not homotopic to any smooth action.
J.-M. Gambaudo and É. Pécou introduced the "linking property" in the study of the dynamics of germs of planar homeomorphisms in order to provide a new proof of Naishul's theorem. In this paper we prove that the negation of the Gambaudo-Pécou property characterizes the topological dynamics of holomorphic parabolic germs. As a consequence, a rotation set for germs of surface homeomorphisms around a fixed point can be defined, and it turns out to be non-trivial except for countably many conjugacy classes....
We describe all the group morphisms from the group of orientation-preserving homeomorphisms of the circle to the group of homeomorphisms of the annulus or of the torus.
Given an orientation-preserving homeomorphism of the plane, a rotation number can be associated with each locally attracting fixed point. Assuming that the homeomorphism is dissipative
and the rotation number vanishes we prove the existence of a second fixed point. The main tools in the proof are Carath´eodory prime ends and fixed point index. The result is applicable to some
concrete problems in the theory of periodic differential equations.
We consider the billiard map in the hypercube of . We obtain a language by coding the billiard map by the faces of the hypercube. We investigate the complexity function of this language. We prove that is the order of magnitude of the complexity.
We study two complex invariant manifolds associated with the parabolic fixed point of
the area-preserving Hénon map. A single formal power series corresponds to both of them.
The Borel transform of the formal series defines an analytic germ. We explore the Riemann
surface and singularities of its analytic continuation. In particular we give a complete
description of the “first” singularity and prove that a constant, which describes the
splitting of the invariant manifolds, does not vanish. An interpretation...
La notion de type géométrique d’une partition de Markov est au centre de la
classification des difféomorphismes de Smale i.e. des difféomorphismes -
structurellement stables des surfaces. On résout ici le problème de réalisabilité : on
donne un critère effectif pour décider si une combinatoire abstraite est, ou n’est pas,
le type géométrique d’une partition de Markov de pièce basique de difféomorphisme de
Smale de surface compacte.
Michael Handel proved the existence of a fixed point for an orientation preserving homeomorphism of the open unit disk that can be extended to the closed disk, provided that it has points whose orbits form an oriented cycle of links at infinity. More recently, the author generalized Handel's theorem to a wider class of cycles of links. In this paper we complete this topic describing exactly which are all the cycles of links forcing the existence of a fixed point.
Un homéomorphisme de Brouwer est un homéomorphisme du plan, sans point fixe, préservant l’orientation. Le théorème des translations planes affirme qu’un tel homéomorphisme s’obtient toujours en « recollant des translations ». Dans cet article, nous introduisons un nouvel invariant de conjugaison des homéomorphismes de Brouwer, l’ensemble oscillant, pour tenter de décrire assez précisément la manière dont s’effectue le recollement des translations.
D’une part, nous utilisons la notion d’ensemble...
Nous définissons la notion d’ensemble bien ordonné de torsion nulle pour les applications déviant la verticale. Contrairement aux études variationnelles de [14] et [1], nous proposons une approche topologique. On retrouve pour ces ensembles un grand nombre de propriétés des ensembles bien ordonnés décrites dans [11]. En reprenant un argument de G.Hall [7], nous montrons en particulier que pour tout nombre de rotation, il existe un ensemble bien ordonné de torsion nulle.
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