Sensitivity of Iterated Function Systems Under the Product Operation

Abstract

In this paper, ${\mathcal {F}}=\{X; f_\lambda |\lambda \in \varLambda _1\}$ and ${\mathcal {G}}=\{Y; g_\gamma |\gamma \in \varLambda _2\}$ are two iterative function systems defined on the compact metric spaces $(X,d_1)$ and $(Y,d_2)$, respectively, where $\varLambda _1,\varLambda _2$ are finite nonempty sets. The concepts of a stronger form of sensitive (i.e., ergodically sensitive, thickly syndetically sensitive, thickly sensitive, syndetically sensitive, cofinitely sensitive, Li–Yorke sensitive, and infinite sensitive) for iterated function systems are introduced. In particular, for every integer $k\ge 2,k\in {\mathbb {N}}$, it is shown that ${\mathcal {F}}\times {\mathcal {G}}$ and ${\mathcal {F}}^{k}\times {\mathcal {G}}^{k}$ are equivalent to Li–Yorke sensitive (resp., infinite sensitive, sensitive). What’s more, some necessary and sufficient conditions for ${\mathcal {F}}\times {\mathcal {G}}$ to be sensitive (resp., ergodically sensitive, thickly syndetically sensitive, thickly sensitive, syndetically sensitive, or cofinitely sensitive) are obtained.