Dynamics of Induced Maps on the Space of Probability Measures

For a continuous self-map f on a compact metric space X, we provide two simple examples: the first confirms that shadowing of (X, f) is not inherited by (M(X) , f^) in general, and the other satisfies that both (X, f) and (M(X) , f^) have no Li–Yorke pair, where M(X) be the space of all Borel probability measures on X. Then we prove that chain transitivity of (X, f) implies chain mixing of (M(X) , f^) , and provide an example to deny the converse. For a non-autonomous system (X, f_,∞) , we prove that weak mixing of (M(X) , f^ _,∞) implies that of (X, f_,∞) , and give an example to deny the converse, where f0,∞={fn}^∞n=0 is a sequence of continuous self-maps on X. We also prove that if f_n is surjective for all n≥ 0 , then chain mixing of (M(X) , f^ _,∞) always holds true, and shadowing of (M(X) , f^ _,∞) implies mixing of (X, f_,∞). If X= I is an interval, we obtain a sharp condition such that transitivity is equivalent between (I, f) and (M(I) , f^). Although (M(I) , f^) has infinite topological entropy for any transitive system (I, f), we give an example such that (I, f_,∞) is transitive but (M(I) , f^ _,∞) has zero topological entropy.