Let $(X,\rho)$ be a Polish space. We are given continuous functions $S_i: X\rightarrow X$ ($i=1,...,N$) and a constant $0
For each $x,y \in X$ there exists $i\leq N$ such that $\rho(S_i(x),S_i(y))\leq r\rho(x,y).$
My question is:
Does there exist a point $x_0$ such that for every $x\in X$ there is a sequence $(i_1,i_2,..)$ such that we have $\lim_{n\to\infty} S_{i_n}\circ...\circ S_{i_1}(x)=x_0\;?$
I think it is too optymistic, however I can not find a suitable counterexample. I would be grateful for some help.