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The Arnold conjecture about fixed points of Hamiltonian diffeomorphisms was partly motivated by the celebrated Poincare-Birkhoff fixed point theorem for an area-preserving homeomorphism of an annulus in the plane. Despite the fact that the Arnold conjecture was formulated in he smooth setting, several attempts to return to the continuous setting of homeomorphisms and to study the conjecture in this setting has been made afterwards. In this talk I will describe some old and more recent results on the subject. Based on a joint work with V. Humiliere and S. Seyfaddini.

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