## Background

Suppose $X$ is a compact metric space, and that $\varphi: X\to X$ is a homeomorphism of $X$.

We say a subset $A$ of $X$ is $\varphi$-invariant if $\varphi(A) = A$. A $\varphi$-invariant set is minimal if it is closed, $\varphi$-invariant, nonempty and the smallest of all such sets. We say $(X,\varphi)$ is essentially minimal if $X$ contains a unique minimal $\varphi$-invariant set.

An orbit of $x \in X$ is the set $O_\varphi(x)= \{ \varphi^n(x) \;|\; n\in\mathbb{Z} \}$.

## My Question

Now suppose that $\varphi: X\to X$ is a homeomorphism such that $X$ contains exactly two minimal $\varphi$-invariant subsets $M_1,M_2\subset X$. Moreover, $\varphi$ has the property that for all $x\in X$ either $M_1\subset\overline{O_\varphi(x)}$ or $M_2\subset\overline{O_\varphi(x)}$ but not both.

Does it follow that $X=X_1\dot{\cup} X_2$ splits into two clopen $\varphi$-invariant subsets?

If not, I would be thankful for a counter example.

More generally, I would like to answer the following more general question. Suppose $\varphi: X\to X$ is a homeomorphism and let $\mathcal{M}$ be the set of all minimal $\varphi$-invariant sets in $X$. Suppose also that $\varphi$ satisfies the following property: $$\forall\; x\in X: \exists !\; M\in\mathcal{M}: M\subset\overline{O_\varphi(x)}. $$

For $M_1,M_2\in\mathcal{M}$, can one find $\varphi$-invariant open subsets $U_1,U_2\subset X$ such that $M_i\subset U_i$ for $i=1,2$ and $U_1\cap U_2=\emptyset$ ?

My motivation is, among other things, to show that certain homeomorphisms with this property can be decomposed into essentially minimal systems, i.e. $X=\bigcup E_\alpha$ and the $E_\alpha$ are closed, $\varphi$-invariant, essentially minimal and pairwise disjoint. For the case I am interested in, a positive answer to the above (more general) question would be sufficient. But since I cannot even answer the more specific question, an answer to that would already be very helpful.