  In this paper, we give the concept of Banach-mean equicontinuity and prove
that three concepts, Bnanach-, Weyl- and Besicovitch-mean equicontinuity of a
dynamic system with abelian group action are equivalent. Furthermore, we obtain
that the topological entropy of a transitive, almost Banach-mean equicontinuous
dynamical system with abelain group action is zero. As an application with our
main result, we show that the topological entropy of the Banach-mean
equicontinuous system under the action of an abelian groups is zero.
