Majorization of invariant measures for orientation-reversing maps

Let the invariant probability measures for an orientation-reversing weakly expanding map of the interval [0, 1] be partially ordered by majorization. The minimal elements of the resulting poset are shown to be convex combinations of Dirac measures supported on two adjacent fixed points. A consequence is that if f : [0, 1] → R is strictly convex, then either its minimizing measure is unique and is a Dirac measure on a fixed point, or f has precisely two ergodic minimizing measures, namely Dirac measures on two adjacent fixed points. In the case where {0, 1} is a period-two orbit, with corresponding invariant measure μ01, the maximal elements of the poset are shown to be convex combinations of μ01 with the Dirac measure on either the leftmost, or the rightmost, fixed point. This facilitates the identification of f -maximizing measures when f : [0, 1] → R is convex.