Abstract:

Let $Asubset R^2$ be an open connected set and $f = (f_1,f_2): A to R^2$ be a map of class $C^2$ with nowhere zero Jacobian determinant in $A$. We define $H: A to R$ by [ H(x,y) = frac{f_1(x,y)^2 + f_2(x,y)^2}{2}. ] It is simple to prove that if $f(z_0) = (0,0)$, then $z_0$ is a center of the Hamiltonian system induced by $H.$ In case $A = R^2$, $f$ is polynomial and $z_0 = (0,0)$, Sabatini proved in 1998 that $z_0$ is a global center if and only if $f$ is a global diffeomorphism. In this talk we recall this result of Sabatini, and give some applications. We will also explain a generalization of Sabatini's result to the context of $C^2$ maps defined in open connected sets.

This talk is based on joint work with J. Llibre.