Fanni M. Sélley

Title: Stability of the invariant distribution in infinite systems of coupled maps

Abstract: In this talk we introduce a family of globally coupled circle maps. Assuming some regularity conditions, we show that for sufficiently weak coupling the system has a unique invariant distribution in a suitable space of (Lebesgue-absolutely continuous) measures. We also show that initial densities close to the unique invariant density converge to it with exponential speed. This might not be the case for sufficiently strong coupling. We show an example where the distributions do not converge, but approach a moving point mass in any sensible metric on spaces of measures. This can be interpreted as the perfect synchronization of the coupled map system. Joint work with Gerhard Keller, Péter Bálint and Imre Péter Tóth.