Attractor and its self-similarities for an IFS over arbitrary sub-shift

Abstract

Consider a compact metric space $X$, and let $\mathcal{F}=\{f_1,\,f_2,\ldots,\, f_k\}$ be a set of contracting and continuous self maps on $X$. Let $\Sigma$ be a sub-shift on $k$ symbols, and let $\Sigma_k$ be the full shift. Define $\mathcal{L}_n(\Sigma)$ as the set of words of length $n$ in $\Sigma$. For $u=u_1\cdots u_n\in \mathcal{L}_n(\Sigma)$, set $f_u:=f_{u_n}\circ\cdots \circ f_{u_1}$ and $H^n(\cdot):=\cup_{u \in \mathcal{L}_{n}(\Sigma_k)} f_{u}(\cdot)$. When $\Sigma=\Sigma_k$, $H^n(\cdot)$ is the $n$th iteration of the Hutchinson's operator, and there exists a compact set $S= \lim_{n \rightarrow \infty} H^n(A)$ for any compact $A\subseteq X$ with $H^n(S)=S$ (self-similarity criteria) for $n\in\N$. For arbitrary $\Sigma$, the above limit exists; but it is not necessarily true that $H^n(S)=S$. Sufficient conditions on $\Sigma$ are provided to have $H^n(S)=S$ for all or some $n\in\N$, and then the dynamics of $S$ under the admissible iterations of $f_i$'s defined by $\Sigma$ are investigated.