Concave unimodal maps have no majorisation relations between their ergodic measures

Steel, Jacob

doi:10.1090/S0002-9939-2010-10705-0

Concave unimodal maps have no majorisation relations between their ergodic measures
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Proc. Amer. Math. Soc. 139 (2011), 2553-2558 Request permission

Abstract:

If $T$ is a concave unimodal map on the unit interval $[0,1]$ and $\{x:T^{n}(x) = 1 \text { for some } n\}$ 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.

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