Concave unimodal maps have no majorisation relations between their ergodic measures

Abstract

If T is a concave unimodal map on the unit interval [0,1] and {x: T n (x) = 1 for some n} is dense in [0,1], we prove that all T-invariant ergodic Borel probability measures are mutually incomparable with respect to the partial order of majorisation. This contrasts sharply with the situation for other interval maps previously studied.