Patrice Le Calvez

Une version feuilletée équivariante du théorème de translation de Brouwer

The Brouwer’s plane translation theorem asserts that for a fixed point free orientation preserving homeomorphism f of the plane, every point belongs to a Brouwer line: a proper topological embedding C of R, disjoint from its image and separating f(C) and f–1(C). Suppose that f commutes with the elements of a discrete group G of orientation preserving homeomorphisms acting freely and properly on the plane. We will construct a G-invariant topological foliation of the plane by Brouwer lines. We apply this result to give simple proofs of previous results about area-preserving homeomorphisms of surfaces and to prove the following theorem: any Hamiltonian homeomorphism of a closed surface of genus g ≥ 1 has infinitely many contractible periodic points.

UNE PROPRIÉTÉ DYNAMIQUE DES HOMÉOMORPHISMES DU PLAN AU VOISINAGE D’UN POINT FIXE D’INDICE >1

Nous montrons qu’un homeomorphisme f du plan preservant l’orientation defini au voisinage d’un point fixe isole d’indice de Lefschetz strictement superieur a 1 admet dans ce voisinage un domaine positivement ou negativement errant. Une telle situation est bien sur impossible quand f preserve l’aire et l’indice ecst donc necessairement inferieur ou egal a 1. We prove that every orientation preserving homeomorphism f of the plane defined locally around an isolated fixed point of Lefschetz index strictly larger than 1 has a positively or negatively wandering domain in this neighbourhood. Such a situation cannot occur when f is area preserving and the index must be smaller or equal to 1.

Rotation numbers in the infinite annulus

Using the notion of free transverse triangulation we prove that the rotation number of a given probability measure invariant by a homeomorphism of the open annulus depends continuously on the homeomorphism under some boundedness conditions.

Drift orbits for families of twist maps of the annulus

We generalize the classical result of J. Mather stating the existence of a drift orbit inside a region of instability of an exact-symplectic positive twist map, to the case of a finite family

Une nouvelle preuve du théorème de point fixe de Handel

{\cal F}

Une généralisation du théorème de Conley–Zehnder aux homéomorphismes du tore de dimension deux

of such maps. A special case is the case where the maps

STABLY NONSYNCHRONIZABLE MAPS OF THE PLANE

F\in{\cal F}

Dynamique des homéomorphismes du plan au voisinage d'un point fixe

have no common invariant continuous graph. We prove the existence of a sequence

Pourquoi les points périodiques des homéomorphismes du plan tournent-ils autour de certains points fixes ?

(z_i)_{i\in\mathbb{Z}}

Propriétés dynamiques des régions d'instabilité

in