Distributional chaos occurring on measure center

In studying a dynamical system, we know that there frequently exist some kinds of disturbance or false phenomenon, namely that some chaotic sets are included in the Borel sets with absolute measure zero. From the viewpoint of ergodic theory, a Borel set with absolute measure zero is negligible. In order to remove these disturbance and false phenomenon, reference Zhou (1993) introduced a concept of measure center. This paper is concerned with distributional chaos occurring on measure center. We draw three conclusions: (i) the one-sided shift has an uncountable distributionally scrambled set which is included in the set of all weakly almost periodic points but not in the set of almost periodic points; (ii) we give a sufficient condition for the compact dynamic system (X,f)(X,f) to exhibit distributional chaos on measure center via semiconjugacy; (iii) the one-sided shift on the measure center is distributionally chaotic in sequence.