Local ergodicity of linear contractions on 𝐶(𝑋)

Abstract

The linear contraction T T on C ( X ) C\left ( X \right ) is locally strongly ergodic if and only if it is continuously scattered and the following condition holds: the support of every extremal (with norm one), T ∗ {T^ * } -invariant measure is contained in the set { x : h ( x ) ≠ 0 } \left \{ {x:h\left ( x \right ) \ne 0} \right \} for some continuous function h h which is T T -invariant on the center.