On almost-periodic points of a topological Markov chain

We prove that a transitive topological Markov chain has almost-periodic points of all $D$-periods. Moreover, every $D$-period is realized by continuously many distinct minimal sets. We give a simple constructive proof of the result which asserts that any transitive topological Markov chain has periodic points of almost all periods, and study the structure of the finite set of positive integers that are not periods.