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A topological characterization of holomorphic parabolic germs in the plane

Frédéric Le Roux (2008)

Fundamenta Mathematicae

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....

Attractors with vanishing rotation number

Rafael Ortega, Francisco Ruiz del Portal (2011)

Journal of the European Mathematical Society

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.

Compacts connexes invariants par une application univalente

Emmanuel Risler (1999)

Fundamenta Mathematicae

Let K be a compact connected subset of cc, not reduced to a point, and F a univalent map in a neighborhood of K such that F(K) = K. This work presents a study and a classification of the dynamics of F in a neighborhood of K. When ℂ K has one or two connected components, it is proved that there is a natural rotation number associated with the dynamics. If this rotation number is irrational, the situation is close to that of “degenerate Siegel disks” or “degenerate Herman rings” studied by R. Pérez-Marco...

Dynamics of circle maps with flat spots

Jacek Graczyk (2010)

Fundamenta Mathematicae

We study a certain class of weakly order preserving, non-invertible circle maps with irrational rotation numbers and exactly one flat interval. For this class of circle maps we explain the geometric and dynamic structure of orbits. In particular, we formulate the so called upper and lower scaling rules which show an asymmetric and double exponential decay of geometry.

Ensembles de torsion nulle des applications déviant la verticale

Sylvain Crovisier (2003)

Bulletin de la Société Mathématique de France

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.

Groups of real analytic diffeomorphisms of the circle with a finite image under the rotation number function

Yoshifumi Matsuda (2009)

Annales de l’institut Fourier

We consider groups of orientation-preserving real analytic diffeomorphisms of the circle which have a finite image under the rotation number function. We show that if such a group is nondiscrete with respect to the C 1 -topology then it has a finite orbit. As a corollary, we show that if such a group has no finite orbit then each of its subgroups contains either a cyclic subgroup of finite index or a nonabelian free subgroup.

Multivalued Lyapunov functions for homeomorphisms of the 2-torus

Patrice Le Calvez (2006)

Fundamenta Mathematicae

Let F be a homeomorphism of 𝕋² = ℝ²/ℤ² isotopic to the identity and f a lift to the universal covering space ℝ². We suppose that κ ∈ H¹(𝕋²,ℝ) is a cohomology class which is positive on the rotation set of f. We prove the existence of a smooth Lyapunov function of f whose derivative lifts a non-vanishing smooth closed form on 𝕋² whose cohomology class is κ.

Nombre de rotation, mesures invariantes et ratio set des homéomorphismes affines par morceaux du cercle

Isabelle Liousse (2005)

Annales de l’institut Fourier

Etant donné α irrationnel de type constant, nous donnons des conditions explicites et génériques sur les pentes d’un homéomorphisme f affine par morceaux du cercle de nombre de rotation α , qui garantissent que la mesure de probabilité f -invariante est singulière par rapport à la mesure de Haar. Cet article contient une preuve élémentaire d’un résultat de E. Ghys et V. Sergiescu : ”le nombre de rotation d’un homéomorphisme dyadique est rationnel”. Nous y étudions aussi le ratio set des homéomorphismes...

On the structure of homeomorphisms of the open annulus

Lucien Guillou (2011)

Annales de la faculté des sciences de Toulouse Mathématiques

Let h be a without fixed point lift to the plane of a homeomorphism of the open annulus isotopic to the identity and without wandering point. We show that h admits a h -invariant dense open set O on which it is conjugate to a translation and we study the action of h on the compactly connected components of the closed and without interior set R 2 ∖ O .

On the topological dynamics and phase-locking renormalization of Lorenz-like maps

Lluis Alsedà, Antonio Falcó (2003)

Annales de l’institut Fourier

The aim of this paper is twofold. First we give a characterization of the set of kneading invariants for the class of Lorenz–like maps considered as a map of the circle of degree one with one discontinuity. In a second step we will consider the subclass of the Lorenz– like maps generated by the class of Lorenz maps in the interval. For this class of maps we give a characterization of the set of renormalizable maps with rotation interval degenerate to a rational number, that is, of phase–locking...

Rigidity of critical circle mappings I

Edson de Faria, Welington de Melo (1999)

Journal of the European Mathematical Society

We prove that two C 3 critical circle maps with the same rotation number in a special set 𝔸 are C 1 + α conjugate for some α > 0 provided their successive renormalizations converge together at an exponential rate in the C 0 sense. The set 𝔸 has full Lebesgue measure and contains all rotation numbers of bounded type. By contrast, we also give examples of C ∞ critical circle maps with the same rotation number that are not C 1 + β conjugate for any β > 0 . The class of rotation numbers for which such examples exist contains...

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