Preservation of ergodicity under perturbation (Q1346346)

Let \(T\), \(T'\) be two arbitrary selfmaps on a metric space \(( , )\) and let for every bounded set \(B\), \(| T|_ B\) and \((T,T')_ B\) denote \(\sup_{f,f'\in B}(fT, f'T)\) and \(\sup_{f\in B}(fT, fT')\), respectively. \(T\) is called bounded (a family \(T_ n\) is called uniformly bounded) if \(| T|_ B< \infty\) (respectively, if \(\sup_ n | T_ n|_ B< \infty\)) for every bounded set \(B\). The authors state the following result which, they remark, entails Mott's theorem \(M\). Theorem. Let \(T_ n\) be eventually uniformly bounded and uniformly equicontinuous, and let \(T_ n\cdots T_{n+p}\) converge in \(p\), uniformly in \(n\), to a constant map \(Z\). Let \((T_ n', T_ n)_ B\to 0\) for every bounded set \(B\) and suppose \(\limsup_ p | T'\cdots T_{n+p}'|_ B< \infty\) for every \(n\) and \(B\). Then \(T_ n'\cdots T_{n+p}'\to Z\) for every \(n\).