Patrice Le Calvez (Paris XIII)

A foliated equivariant version of Brouwer's Plane Translation theorem and some applications

Abstract. Let G be a discrete group of orientation preserving homeomorphisms acting freely on the plane. If f is an orientation preserving homeomorphism which commutes with the elements of the group and which is fixed point free., one may construct a topological foliation of the plane which is G-invariant and such that every leaf is a Brouwer line (it separates its image and preimage by f). This equivariant foliated version of the Brouwer plane translation theorem has some applications to the study of area preserving homeomorphisms of surface. We will present one of them: if M is a closed surface of genus $g\geq 1$ and F a hamiltonain homeomorphism (that means the time one map of a 1-periodic time dependent hamiltonian vector field), then F has an infinte number of periodic points corresponding to contractile periodic orbits of the vector field. The other ingedients in the proof are more classical objets of 2-dimentional dynamics: dynamics of foliations, Conley index for discrete maps, prime end theory, topological versions of the Poincaré-Birkhoff theorem.