Geometric Barycentres Of Invariant Measures For Circle Maps

For a continuous circle map T, deene the barycentre of any T-invariant probability measure to be b() = R S 1 z dd(z). The set of all such barycentres is a compact convex subset of C. If T is conjugate to a rational rotation via a MM obius map, we prove 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 maximises 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((), 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. Let T : S 1 ! S 1 be a continuous map of the circle. The set M of T-invariant Borel probability measures is non-empty, by the Krylov-Bogolioubov Theorem. Geometrically M is a simplex, and its extremal points are precisely the ergodic measures. For many circle maps M has a rich structure, and is hard to describe properly. In particular it may well be innnite-dimensional, allowing rather counter-intuitive properties such as density of extremal points (in which case M is the so-called Poulsen simplex, see Lindenstrauss et al. 12], Glasner & Weiss 8]). We hope to gain insight into M by considering nite dimensional projections. In particular, to each measure 2 M we deene its barycentre (or rst Fourier coeecient) to be b() = R S 1 z dd(z), the point in the plane describing the average position of-almost every orbit of T. This is geometrically the most natural two-dimensional projection. We deene = b(M) to be the set of all such barycentres. If w 2 then we let M(w) denote the convex set b ?1 (w) M. We will study the geometry of , and its relation to M. For certain well-known maps the boundary @ has rather pathological diierentiability properties (see Bousch 2], Hunt & Ott 10], Jenkinson 11], and also x7, x8). In this article we mainly focus on the interior of. We highlight a two-parameter family …