Mark Pollicott: (Non)-Abelian Livsic Theorems

Abstract:  In 1972, Livsic proved his famous theorem for Anosov systems that when a Holder continuous real valued function has zero averages around all closed orbits then the function necessarily takes a special form (i.e., it is a coboundary). This is a simple, but striking, example of where local information (on closed orbits) gives a global conclusion (the function is a coboundary). Recently, A. Gogolev and F. Rodriguez Hertz showed that knowing this property for a suitable smaller collection of closed orbits still gives that the Holder function takes a special form (The Abelian Livsic Theorem). We will show a generalization of this result. This is joint work with Richard Sharp (Warwick)