Ruelle Operator for Infinite Conformal IFS

Let $(X,\{w_j{\}}_{j=1}^m,\{p_j{\}}_{j=1}^m)$ ($2\le m<\mathrm{\infty }$) be a contractive iterated function system (IFS), where $X$ is a compact subset of ${\mathbb{R}}^d$. It is well known that there exists a unique nonempty compact set $K$ such that $K={\bigcup }_{j=1}^m{w}_j(K)$. Moreover, the Ruelle operator on $C(K)$ determined by the IFS $(X,\{w_j{\}}_{j=1}^m,\{p_j{\}}_{j=1}^m)$ ($2 \leq m < \infty$) has been introduced in \cite{FL}. In the present paper, the Ruelle operators determined by the infinite conformal IFSs are discussed. Some separation properties for the infinite conformal IFSs are investigated by using the Ruelle operator.