Dynamical Resistance Day

Title: Periodic orbit growth on covers of Anosov flows (video)

It is well-known that the topological entropy of an Anosov flow on a compact manifold describes the exponential growth rate of its periodic orbits. If we pass to a regular cover of the manifold then we can consider a corresponding growth rate for periodic orbits of the lifted flow. This growth rate is bounded above by the original topological entropy but if the cover is infinite then the growth rate may be strictly smaller. In the important special case of a geodesic flow over a compact manifold with negative sectional curvatures, we have equality if and only if the cover is amenable. This statement fails for general Anosov flows but we will discuss a recent result that gives a natural generalisation. This is joint work with Rhiannon Dougall.