This is closely related to a previous question I recently asked. This was about the definition of a locally lifted measure described by Keane in his seminal paper on $g$-measures [1]. I received a good answer, but another way of constructing a measure is through the construction of a particular partition, which I mention in the other post. For convenience I repeat the essentials:
The setting is as follows: Let $(X,T)$ be a dynamical system, where $X$ is a compact metric space $(X,d)$, and $T$ is a minimal covering transformation. By this we (read Keane) mean that:
- $T$ is everywhere $n$-to-1 ($n\geq 2$).
- $T$ is a local homeomorphism.
- There exists a $C>1$, such that for all $x\in X$ there exists a $\delta_x>0$, such that $d(x,y)<\delta_x\implies d(Tx,Ty)\geq Cd(x,y)$.
- For all $\varepsilon>0$ there exists a $N\in\mathbb N$, such that $T^{-N}(x)$ is $\varepsilon$ dense in $X$, for all $x$.
A typical example of covering transformations are the expanding circle endomorphisms, $T_n:S^1\to S^1$, given by $T_ne^{i\theta}=e^{in\theta}$. For these examples we can partition $S^1$ into the disjoint sets $\{X_i\}_{i=1}^n$, where each $X_i=\left[\frac{2\pi(i-1)}{n},\frac{2\pi i}{n}\right)$. These partitions are important, because for each $i$ the restricted map $T|_{X_i}:X_i\to S^1$ is a homeomorphism. This motivates my question:
For the general case, with $X$ an arbitrary compact metric space and $T$ an arbitrary covering transformation satisfying points 1) through 4) above, does a covering transformation always induce a disjoint partition $\{X_i\}_{i=1}^n$ of $X$ such that each restricted map $T|_{X_i}$ is a homeomorphism? Importantly I require the number of sets in the partition, $n$, to equal the number of covers of the covering transformation.
My instinct is yes, but I am not sure how to rigorously construct such a partition. My idea would be to use choice somehow to order the $n$ elements of each $T^{-1}x$ such that we can take $X_i=\bigcup_{x\in X} x_i$, where $x_i$ is the $i$-th element in $T^{-1}x$ according our ordering. I am not sure how to rigorously construct such an ordering though, and would appreciate any input.
[1] Keane, Michael, Strongly mixing g-measures, Invent. Math. 16, 309-324 (1972). ZBL0241.28014.
Assume that $X$ is compact (not necessarily metric) and $T : X \to X$ is a local homeomorphism. Then each $x \in X$ has an open neighborhood $U_x$ which is mapped by $T$ homeomorphically onto $T(U_x)$. By compactness there exist $x_1,\dots,x_n$ such that the $U_i = U_{x_i}$ cover $X$. Now define $$X_i = U_i \setminus \bigcup_{j=1}^{i-1}U_j \subset U_i .$$