Topology and Dynamics of Solenoids of Split Sequences

Sarasi Jayasekara (Rutgers)

Abstract: Given a space of dynamical systems (such as the measured foliation space of a compact surface $S$), and a group $G$ that "nicely" acts on that space (such as the mapping class group of $S$), we can extract dynamical information about the group $G$, by observing the aforementioned space of dynamical systems. There have been many attempts at adopting this point of view to study the group of outer automorphisms of a free group (denoted $Out(F_n)$). To make partial strides towards studying the dynamics of $Out(F_n)$, we introduce a class of objects generalizing the concept of toral solenoids, called "Solenoids of Split Sequences", defined as the inverse limit object of a sequence of inverse folds. We show that solenoids have a natural singular foliation structure, provide an atlas to observe this foliation, and construct a machinery for studying transverse measures on them. We show that the space of transverse measures of an "expanding" solenoid is finite dimensional, and we provide two combinatorial criteria that let us recognize wide classes of topologically minimal solenoids and uniquely ergodic solenoids.