A systems $<f\_0,f\_1>$ is minimal if the set $\{h(x): h=f\_{i\_n}\circ f\_{i\_{n-1}}\circ...\circ f\_{i\_1}, i\_k \in \{0,1\},n>0\}$ is dense in $S^1$, for every $x\in S^1$.
Consider $f:S^1 \to S^1, f(x)=2x (mod \, 1)$ and $f\_{\epsilon}=f+ \epsilon$.
Is there an $\epsilon>0$ such that $<f,f\_{\epsilon}>$ is minimal?



IFS maps on circle
Yes, for example, you could choose $\epsilon$ so that $\{2^k \epsilon \mid k = 1,2,3,\ldots\}$ is dense in $S^1$. By letting all but one of the $f\_{i\_k}$'s be $f$, we see that the orbit of $x$ under words of length $n$ contains $f^n(x) + 2^k\epsilon$ for $1 \le k \le n$.


