Laboratoire de Mathématiques Raphaël Salem

UMR 6085 CNRS - Université de Rouen Normandie

I will introduce certain class of infinite measure preserving dynamical systems, called skew products. In general, it is very difficult to study ergodic properties of such maps. One of the very few tools that we have, are so called essential values. After presenting this notion, I will focus on the main result that I obtained with Frank Trujillo, which says that for a typical symmetric interval exchange transformation, a skew product given by the indicator of the interval $(0,1/2)$ is ergodic. It is the first result of this kind when the cocycle is fixed and the number of exchanged intervals is arbitrary. Time permitting, I will show some tools to obtain this  result. In particular, I will show how to bound Birkhoff sums of certain points, when the function is odd on the interval.