Recurrence and topological entropy of translation operators

In this paper, we provide several characterizations on weak recurrence of translation operators on weighted Lebesgue spaces and on continuous function spaces. It is shown that the translation operator admits a nonzero nonwandering point if and only if it has a nonzero recurrent point, if and only if it is hypercyclic; that the translation operator admits a nonzero almost periodic point if and only if it is Devaney chaotic. Moreover, every hypercyclic translation operator has infinite topological entropy, but the converse is not true. Finally, we remark that there exists a linear operator which has the specification property on the whole space.