Geometric barycentres of invariant measures for circle maps

Abstract

For a continuous circle map T, define the barycentre of any T-invariant probability measure \mu to be b(\mu)=\int_{S^1} z\, d\mu(z). The set \Omega of all such barycentres is a compact convex subset of \mathbb{C}. If T is conjugate to a rational rotation via a Möbius map, we prove \Omega is a disc. For every piecewise-onto expanding map we prove that the barycentre set has non-empty interior. In this case, each interior point is the barycentre of many invariant measures, but we prove that amongst these there is a unique one which maximizes entropy, and that this measure belongs to a distinguished two-parameter family of equilibrium states. This family induces a real-analytic radial foliation of int(\Omega), centred around the barycentre of the global measure of maximal entropy, where each ray is the barycentre locus of some one-parameter section of the family. We explicitly compute these rays for two examples. While developing this framework we also answer a conjecture of Z. Coelho [6] regarding limits of sequences of equilibrium states.