Is every measure-theoretic factor map between isomorphic odometers an isomorphism?

Question

There are odometers that "look different" but are isomorphic, e.g., the odometer (4, 16, 64, ...) and the odometer (2, 4, 8, ...). In the case above, the natural factor map ends up being an isomorphism. Is this true for every factor map between isomorphic odometers?

Answer 1

This is true, and follows from the property called coalescence.

In a topological setting this definition is due to Auslander [Endomorphisms of minimal sets, Duke Math. J. Volume 30, Number 4 (1963), pp. 605-614]. In order to answer your question it is enough to state Auslander's results as follows:

Let $(X,T)$ be a minimal compact dynamical system (that is, $X$ is a compact metric space and $T\colon X\to X$ is a continuous map without nonempty proper closed invariant sets). Any odometer is a minimal compact dynamical system.

A continuous onto map $\varphi\colon X\to X$ such that $\varphi\circ T=T\circ \varphi$ is an endomorphism. An automorphism is an injective endomorphism.

A dynamical system $(X,T)$ is coalescent if every endomorphism is an automorphism of the system. Auslander proved that (see Corollary 3 in his paper) that almost periodic (that is, equicontinuous) minimal sets are coalescent. It is well known that all odometers are equicontinuous. Now assume that you have two isomorphic odometers $(X,T)$ and $(Y,S)$, and a factor map $\varphi\colon X\to Y$. Let $\psi\colon Y\to X$ be an isomorphism. Then $\psi\circ\varphi$ is an endomorphism of $X$, which by coalescence must be invertible, which in turn implies that $\varphi$ is an isomorphism.

There is a corresponding notion of coalescent measure-preserving systems, and odometers are known to be coalescent also in this sense, see F.J. Hahn, W. Parry, Some characteristic properties of dynamical systems with quasi-discrete spectra Math. Systems Theory 2 (1968), pp. 179-190.

Answer 2

According to Theorem 1.3 of Survey of odometers and Toeplitz flows, the kernel of a topological group factor map from $G_{\mathbf{k}}$ to $G_{\mathbf{k'}}$ is $G_{\mathbf{k}-\mathbf{k'}}$. Here $G_{\mathbf{k}}$ is the odometer corresponding to the supernatural number $\prod_{i \in \mathbb{N}^+} \text{p}_i^{k_i}$ with exponents $\mathbf{k}=(k_1,k_2,k_3,\dots)$, i.e. it is the inverse limit of the system $\mathbb{Z}_{a_1} \leftarrow \mathbb{Z}_{a_2} \leftarrow \dots$ with $a_i | a_{i+1}$ and $\text{lcm}(a_i)=\prod_{i \in \mathbb{N}^+} \text{p}_i^{k_i}$. Moreover, two odometers are isomorphic as topological groups if and only if their associated supernatural numbers are the same.

Consequently, any topological group factor map between isomorphic odometers has trivial kernel. One can also prove this for dynamical system factor maps by turning these into topological group factor maps via rotation by the image of $\mathbf{0}$ (or, you can check Lemma 11 of this paper). It turns out that any measure-theoretic factor map between odometers (equipped with the Haar measure and the transformation $w \mapsto w + \mathbf{1}$) is equal to a dynamical system factor map on a full measure set. See Proposition 10 of this paper for a proof of this result. Thus any measure-theoretic factor map on an odometer is injective on a full measure set.