  Motivated by recent investigations \cite{Costakis, Bonilla} on the notion of
recurrence in linear dynamics, we deepen into the notions of recurrence and
frequent recurrence in the setting of dissipative composition operators with
bounded distortion, a class of linear operators which includes backward shifts.
Among other results, we show that these two notions are, actually, equivalent
to those of hypercyclicity and frequent hypercyclicity, respectively. More
particularly, we improve \cite[Theorem 5.2]{Bonilla} and, also, provide an
answer to an open question posed in \cite[Question 5.3]{Bonilla}.
