Poincaré-birkhoff Fixed Point Theorem and Periodic Solutions of Asymptotically Linear Planar Hamiltonian Systems

This work, which has a self contained expository character, is devoted to the Poincaré-Birkhoff (PB) theorem and to its applications to the search of periodic solutions of nonautonomous periodic planar Hamiltonian systems. After some historical remarks, we recall the classical proof of the PB theorem as exposed by Brown and Neumann. Then, a variant of the PB theorem is considered, which enables, together with the classical version, to obtain multiplicity results for asymptotically linear planar hamiltonian systems in terms of the gap between the Maslov indices of the linearizations at zero and at infinity. 1. The Poincaré-Birkhoff theorem in the literature In his paper [28], Poincaré conjectured, and proved in some special cases, that an areapreserving homeomorphism from an annulus onto itself admits (at least) two fixed points when some “twist” condition is satisfied. Roughly speaking, the twist condition consists in rotating the two boundary circles in opposite angular directions. This concept will be made precise in what follows. Subsequently, in 1913, Birkhoff [4] published a complete proof of the existence of at least one fixed point but he made a mistake in deducing the existence of a second one from a remark of Poincaré in [28]. Such a remark guarantees that the sum of the indices of fixed points is zero. In particular, it implies the existence of a second fixed point in the case that the first one has a nonzero index. In 1925 Birkhoff not only corrected his error, but he also weakened the hypothesis about the invariance of the annulus under the homeomorphism T . In fact Birkhoff himself already searched a version of the theorem more convenient for the applications. He also generalized the area-preserving condition. Before going on with the history of the theorem we give a precise statement of the classical version of Poincaré-Birkhoff fixed point theorem and make some remarks. In the following we denote by A the annulus A := {(x, y) ∈ R2 : r2 1 ≤ x2 + y2 ≤ r2 2 , 0 < r1 < r2} and by C1 and C2 its inner and outer boundaries, respectively. ∗The second author wishes to thank Professor Anna Capietto and the University of Turin for the invitation and the kind hospitality during the Third Turin Fortnight on Nonlinear Analysis.