Dynamics of Induced Maps on the Space of Probability Measures

Induced dynamics of non-autonomous dynamical systems

“…Remark 2.11. This result also holds for the induced system on the space M(X) of all Borel probability measures, see Theorem 5.4 in [25].…”

“…) is said to be chain transitive if for any x, y ∈ X and any ǫ > 0, there exists an ǫ-chain from x to y; chain weakly mixing of some order n if (X n , f (n) 0,∞ ) is chain transitive; chain weakly mixing of all orders if (X n , f (n) 0,∞ ) is chain transitive for all n ∈ Z + ; chain mixing if for any ǫ > 0 and x, y ∈ X, there exists N ∈ Z + such that for any n ≥ N , there exists an ǫ-chain from x to y with length n. It follows from Lemma 3.1 in [25] that chain mixing⇒chain weakly mixing of all orders ⇒chain transitivity for (X, f 0,∞ ). Theorem 3.1.…”

“…Chain mixing of (M (X), f0,∞ ) does not imply that of (X, f 0,∞ ), even for a single map, see Example 3.7 in [25]. However, it is not the case for (K(X), f0,∞ ) and (F 1 (X), f0,∞ ).…”

“…Remark 4.3. Example 3.9 in [25] shows that shadowing of (X, f 0,∞ ) is not inherited by (M (X), f0,∞ ) in general. This is another difference of dynamics between (K(X), f0,∞ ) and that of (M (X), f0,∞ ).…”

“…They also studied the other induced system (M (X), f ) on the space M (X) consisting of all Borel probability measures with the Prohorov metric and f is the induced map on M (X). The readers are referred to [3,6,11,17,18,19,25] and references therein for the dynamics of (M (X), f ). It is worthy to mention that Glasner and Weiss [11] proved that (X, f ) has zero topological entropy if and only if (M (X), f ) has zero topological entropy, which demonstrates a big difference of dynamics between (K(X), f ) and (M (X), f ).…”