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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 $0<r<1$ such that $$(*) \quad \sum_{i=1}^N p_i(x) \rho(S_i(x), S_i(y))\leq r \rho(x,y)\;\;\; (x,y \in X)$$ and $\inf_x p_i(x)>0$, $i=1,...,N$. Clearly, the transition operator of our IFS is given by: $$\mu P=\sum_{i=1}^N \int_{X} 1_A(S_i(x))p_i(x)\,\mu(dx)\;\;\; (\mu\in\mathcal{M}_{prob}(X), \; A\in\mathcal{B}(X))$$

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 $\mbox{dist } (\mbox{supp } \pi_x, \mbox{supp }\pi_y)=0$

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