Constant-length substitution systems and countable distributionally scrambled sets

In this paper we study the compact dynamical systems which are on the edge of distributional chaos, that is, whose distributionally scrambled sets are finite or countable. First we show that a constant-length substitution system without eventually periodic substitution sequence has only finite distributionally scrambled sets. Then we give some constant-length substitution systems whose distributionally scrambled sets may have any given finite cardinal number. At last we provide a compact dynamical system generated by some constant-length substitution systems, whose distributionally scrambled set has at most countably many elements.