# From Brouwer theory to the study of homeomorphisms of surfaces

### Patrice Le Calvez

Université de Paris XIII, Villetaneuse, France

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## Abstract

We will state an equivariant foliated version of the classical Brouwer Plane Translation Theorem and will explain how to apply this result to the study of homeomorphisms of surfaces. In particular we will explain why a diffeomorphism of a closed oriented surface of genus ≥ 1 that is the time-one map of a time dependent Hamiltonian vector ﬁeld has inﬁnitely many periodic orbits. This gives a positive answer in the case of surfaces to a more general question stated by C. Conley. We will give a survey of some recent results on homeomorphisms and diffeomorphisms of surfaces and will explain the links with the improved version of Brouwer’s theorem.