A quantitative approach to syndetic transitivity and topological ergodicity

In this paper, we give new quantitative characteristics of degrees of syndetical transitivity and topological ergodicity for a given discrete dynamical system, which are nonnegative real numbers and are not more than 1. For selfmaps of many compact metric spaces it is proved that a given selfmap is syndetically transitive if and only if its degree of syndetical transitivity is 1, and that it is topologically ergodic if and only if its degree of topological ergodicity is one. Moreover, there exists a selfmap of [0, 1] having all degrees positive.