Check the following. In the proof of the factorization lemma (Lemma 8.14 ) we cannot, in general, replace \(\liminf _{n} w_{n}\) by \(\lim _{n} w_{n}\).

[ find a sequence \(\left(w_{n}ight)_{n}\) and a map \(T\) such that \(\left(w_{n} \circ Tight)_{n}\) is convergent and \(\left(w_{n}ight)_{n}\) divergent.]