Let $(X, \rho)$ be a Polish space. Consider an Iterated Function System $(S_i,p_i)_{i=1,...,N}$, where $S_i:X\rightarrow X$, $p_i: X\rightarrow \left[0,1\right]$ are continuous functions and $\sum_{i=1}^N p_i(x)=1$, for all $x\in X$. We assume that there exists $0
My question is the following:
Assume that $\delta_xP^n\stackrel{w}{\rightarrow}\pi_x$ and $\delta_yP^n\stackrel{w}{\rightarrow}\pi_y$ (as usual $P^n$ is determined by the Chapman-Kolmogorov equation), where $\pi_x$ and $\pi_y$ are invariant distributions of $P$. Is true that $\mbox{supp } \pi_x \cap \mbox{supp }\pi_y\neq \emptyset$?
It seems to be true because one can show that $\mbox{supp }\delta_x P^n=\{(S_{i_n}\circ ...S_{i_1})(x): i_1,..,i_n\in\{1...,N\}\}$ and from (*) it follows that for any $x,y \in X$ we can choose $i_1,...,i_m$ such that $\rho((S_{i_m}\circ ...S_{i_1})(x),(S_{i_m}\circ ...S_{i_1})(y))\leq r^m\rho(x,y).$
However, I was able to show only that $\mbox{dist } (\mbox{supp } \pi_x, \mbox{supp }\pi_y)=0$