P-chaos implies distributional chaos and chaos in the sense of Devaney with positive topological entropy

Let f be a continuous map from a compact metric space X to itself. The map f is called to be P-chaotic if it has the pseudo-orbit-tracing property and the closure of the set of all periodic points for f is equal to X. We show that every P-chaotic map from a continuum to itself is chaotic in the sense of Devaney and exhibits distributional chaos of type 1 with positive topological entropy.