Topological dynamics of continuous maps induced on the space of probability measures

Let $f$ be a continuous self-map on a compact interval $I$ and $\hat f$ be the induced map 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 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 the converse may not be true. We prove that if $f_n$ is surjective for all $n\geq 0$, 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 topological mixing of $(X, f_{0,\infty})$. In addition, we prove that topological mixing (resp., mild mixing and topological exactness) of $(X, f_{0,\infty})$ is equivalent to that of $(\mathcal{M}(X),\hat f_{0,\infty})$, and that $(X, f_{0,\infty})$ is cofinitely sensitive if and only if $(\mathcal{M}(X),\hat f_{0,\infty})$ is cofinitely sensitive.