The construction of chaotic maps in the sense of Devaney on dendrites which commute to continuous maps on the unit interval

Let f be a continuous map from the unit interval to itself. In this paper, it is shown that f has positive topological entropy if and only if f is pointwise P-expansive for some periodic orbit P of f. And it is also proved that if f has a periodic orbit with odd period, then there exists a chaotic map from a dendrite to itself in the sense of Devaney which is semiconjugate to f and has positive topological entropy.