I hope you can give me some suggestion.
Definition: Given two bijections $f,g:\mathbb{S}^1\to \mathbb{S}^1$ and $\mathcal{P}$ a partition of $\mathbb{S}^1$ (a partition of $\mathbb{S}^1$ is a finite disjoint collection of nonempty subsets whose union is $\mathbb{S}^1$). We say that $f$ and $g$ are $\mathcal{P}$-close if for every $x\in \mathbb{S}^1$ the set $\{f(x),g(x)\}$ is contained in a element of $\mathcal{P}$.
Let $R:\mathbb{S}^1\to \mathbb{S}^1$ be a rotation such that $R(x)=(x+\frac{1}{\pi}) \mod\ 1$.
I would like to know if $R$ satisfies the following property: Given a partition $\mathcal{P}$, there exists a partition $\mathcal{Q}$ such that if $g:\mathbb{S}^1\to \mathbb{S}^1$ is a bijection $\mathcal{Q}$-close with $R$ then there is a map $h:\mathbb{S}^1\to \mathbb{S}^1$ such that
- $h\circ g=R\circ h$
- $h$ is $\mathcal{P}$-close with the identity map $I$ in $\mathbb{S}^1$.
Note: The identity map $I$ satisfies that property. In fact, given $\mathcal{P}=\{A_1,\cdots,A_k\}$ is suffcient to take $\mathcal{Q}=\mathcal{P}$. Let $g:\mathbb{S}^1\to \mathbb{S}^1$ be a bijection $\mathcal{Q}$-close with $I$. Fix a finite collection $\{a_1,\cdots,a_k\}$ where $a_i\in A_i$. Hence the map $h:\mathbb{S}^1\to \mathbb{S}^1$ such that $h(y)=g(a_i)$ whenever $y\in A_i$ satisfies: $h\circ g=h$ and it is $\mathcal{P}$-close with $I$.