Michal Misiurewicz: Concavity implies attraction

We consider a skew product with the interval [0,a] as a fiber space and maps in fibers that are concave and fix 0. If the map in the base is an irrational rotation of a circle, then it has been known that under some additional conditions there exists a Strange Nonchaotic Attractor (SNA) for the system. The proofs involved Lyapunov exponents and Ergodic Theorem. We show that the existence of an attractor basically follows only from the uniform concavity of the maps in the fibers. In particular, it does not depend on the map in the base, so it occurs also in a nonautonomous case. Next, we discuss the possible generalizations of the notion of a SNA and show the problems that can occur in the case when the map in the base is noninvertible. This is a joint work with Lluis Alseda.