On a connection between global centers and global injectivity [ Back ]

Date:
18.04.16   
Times:
15:30
Place:
UAB - Dept. Matemàtiques (C1/-128)
Speaker:
Francisco Braun
University:
Universidade Federal de São Carlos

Abstract:

Let $A\subset \R^2$ 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.