Background
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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 $O\_\varphi(x)= \{ \varphi^n(x) \;|\; n\in\mathbb{Z} \}$.




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




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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 $U\_1\cap U\_2=\emptyset$ ?
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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. 



When does a homeomorphism split into essentially minimal homeomorphisms?
The answer is no.


For each pair $(n,k) \in \mathbb{Z}^+ \times \mathbb{Z}$ we define a point $p(n,k) \in \mathbb{R^2}$ as: $$p(n,k)=\left(1+\frac{1}{n},\frac{k}{n^2} \right)$$ if $|k| \leq n$, and $$p(n,k)=\left( \frac{1}{k}\cos(1/n), \frac{1}{k} \sin(1/n) \right)$$ otherwise.


We let $$X= \left\{ p(n,k): (n,k) \in \mathbb{Z}^+ \times \mathbb{Z} \right\} \cup \left\{(0,0), (1,0) \right\}$$ viewed as a (compact) subspace of $\mathbb{R}^2$, and define $\varphi:X \to X$ by $\varphi(0,0)=(0,0)$, $\varphi(1,0)=(1,0)$ and $\varphi(p(n,k))=p(n,k+1)$.


It is easy to check that all the requirements are met (note that the two minimal $\varphi$-invariant sets are $M\_1=\{(0,0)\} $ and $M\_2=\{(1,0)\}$). However, if $U$ is an open subset of $X$ containing $(0,0)$ then for each $n$ there is a $k$ such that $p(n,k) \in U$; so if $U$ is also $\varphi$-invariant then it must contain every $p(n,k)$; therefore, if $U$ is also closed, it must contain $(1,0)$. In other words, the only $\varphi$-invariant clopen is $X$.


