Absolutely continuous invariant measures that are maximal

Let $A$ be a certain irreducible $0{\text {-}}1$ matrix and let $\tau$ denote the family of piecewise linear Markov maps on $[0,1]$ which are consistent with $A$. The main result of this paper characterizes those maps in $\tau$ whose (unique) absolutely continuous invariant measure is maximal, and proves that for "most" of the maps that are consistent with $A$, the absolutely continuous invariant measure is not maximal.