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, whereM(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(0,infinity)), we prove that weak mixing of (M( X), f<^>(0,infinity)) implies that of ( X, f(0,infinity)), and give an example to deny the converse, where f(0,infinity) = {f(n)}8 n=0 is a sequence of continuous self-maps on X. We also prove that if fn is surjective for all n = 0, then chain mixing of (M( X), f<^>(0,infinity)) always holds true, and shadowing of (M( X), f<^>(0,infinity)) implies mixing of ( X, f(0,infinity)). 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(0,infinity)) is transitive but (M( I), f<^>(0,infinity)) has zero topological entropy.