Majorisation ordering of measures invariant under transformations of the interval

Abstract

Majorisation is a partial ordering that can be applied to the set of probability measures on the unit interval I = [0, 1). Its defining property is that one measure μ majorises another measure , written μ , if R I fdμ R I fd for every convex real-valued function f : I ! R. This means that studying the majorisation of MT , the set of measures invariant under a transformation T : I ! I, can give us insight into finding the maximising and minimising T-invariant measures for convex and concave f. In this thesis I look at the majorisation ordering of MT for four categories of transformations T: concave unimodal maps, the doubling map T : x 7! 2x (mod 1), the family of shifted doubling maps T : x 7! 2x + (mod 1), and the family of orientation-reversing weakly-expanding maps.