Let $0<r<1$ and $\alpha\in [0, r]\cup \{1\}$. Take $f_{\alpha}:\mathbb{R}\to \mathbb{R}$ defined by $f_\alpha(x)=\alpha x$. Let $$T=\{f_\alpha: \alpha\in ([0, r]\cup \{1\})\}$$ Is there a Folner sequence for $T$?
A sequence $\{A_n\}_{n\in D}$, where $A_n\subseteq T$ is finite set, is called left folner sequence if and only if for each $t\in T$, the net $$ \frac{|tA_n\Delta A_n|}{|A_n|}$$ Converges to $0$.(Definition 4.1 in Density in Arbitrary Semigroups by Hindman and Strauss.)