Let $(X, \rho)$ be a complete metric space and let $S_1,...,S_N: X\rightarrow X$ be continous functions.
A nonempty compact set $C$ is said to be an attractor for the family $\{S_1,...,S_n\}$ if $$\bigcup_{i=1}^N S_i(C)=C.$$ It is well known that in the case when all the functions $S_i$ are contractions then there exists an attractor for the family $\{S_1,...,S_N\}$ (it can be easily proved by using the Hausdorff distance).
Assume that $$\tag{1} \rho(S_i(x), S_i(y))\leq r(\rho(x,y))\;\;\; (i=1,..,N,\;x,y\in X),$$ where $r:\left[0,\infty\right )\rightarrow \left[0,\infty\right )$ is a nondecreasing and continuous at $0$ function such that $r(0)=0$ and $r^n(t)\to 0$, for all $t\geq 0$.
My question is the following:
Is there exist an attractor for the family $\{S_1,..,S_N\}$ if condition $(1)$ holds?
Denote $$\Gamma^{\infty}(x)=\{S_{i_n}\circ... S_{i_1}(x):\;i_1,...,i_n\in\{1,..,N\},\;n\in\mathbb{N}\}$$
Observe that if $\Gamma$ stands for the Barnsley operator, i.e $\Gamma(A)=\bigcup_{i=1}^N S_i(A)$, then we may write $\Gamma^{\infty}(x)=\bigcup_{n=1}^{\infty}\Gamma^n(\{x\})$. Furthermore, it is easily seen that if $\text{cl } \Gamma^{\infty}(x)$ is compact then it is an attractor for $\{S_1,...,S_N\}$. I have proved that the following conditions are equivalent:
(i) $\text{cl }\Gamma^\infty (x)$ is compact for some $x\in X$,
(ii) $\text{cl }\Gamma^\infty (x)$ is compact for all $x\in X$,
(iii) There exists an attractor for $\{S_1,...,S_N\}$.
So, my question can also be formulate as follows:
Is the set $\text{cl }\Gamma^\infty (x)$ compact if condition $(1)$ holds?
I will be greatfull for any hints.
