Dynamics of continuous maps induced on the space of probability measures

Abstract

For a continuous self-map $f$ on a compact interval $I$ and the induced map $\hat{f}$ on the space $\mathcal{M}(I)$ of probability measures, we obtain a sharp condition to guarantee that $(I,f)$ is transitive if and only if $(\mathcal{M}(I),\hat{f})$ is transitive. We also show that the sensitivity of $(I,f)$ is equivalent to that of $(\mathcal{M}(I),\hat{f})$. We prove that $(\mathcal{M}(I),\hat{f})$ must have infinite topological entropy for any transitive system $(I,f)$, while there exists a transitive non-autonomous system $(I,f_{0,\infty })$ such that $(\mathcal{M}(I),{\hat{f}}_{0,\infty })$ has zero topological entropy, where $f_{0,\infty }=\{f_n{\}}_{n=0}^{\infty }$ is a sequence of continuous self-maps on $I$. For a continuous self-map $f$ on a general compact metric space $X$, we show that chain transitivity of $(X,f)$ implies chain mixing of $(\mathcal{M}(X),\hat{f})$, and we provide two counterexamples to demonstrate that the converse is not true. We confirm that shadowing of $(X,f)$ is not inherited by $(\mathcal{M}(X),\hat{f})$ in general. For a non-autonomous system $(X,f_{0,\infty })$, we prove that if $(\mathcal{M}(X),{\hat{f}}_{0,\infty })$ is weak mixing of order $n$, then so is $(X,f_{0,\infty })$ for any $n\ge 2$; while there exists $(X,f_{0,\infty })$ such that it is weak mixing of order $2$ but $(\mathcal{M}(X),{\hat{f}}_{0,\infty })$ is not. We then prove that Li-Yorke chaos (resp., distributional chaos) of $(X,f_{0,\infty })$ carries over to $(\mathcal{M}(X),{\hat{f}}_{0,\infty })$, and give an example to show that $(X,f)$ and $(\mathcal{M}(X),\hat{f})$ may have no Li-Yorke pair simultaneously. We also prove that if $f_n$ is surjective for all $n\ge 0$, then chain mixing of $(\mathcal{M}(X),{\hat{f}}_{0,\infty })$ always holds true, and shadowing of $(\mathcal{M}(X),{\hat{f}}_{0,\infty })$ implies mixing of $(X,f_{0,\infty })$.

26 pages. Example 5.6 is added