Is there an invertible measure-preserving transformation (preferably a nice one) admitting every irrational rotation as a factor ? I guess the spectrum is the relevant tool to address this question but I do not master spectral theory. Of course, if it is true, I am interested in a "minimal" such transformation. It is understood I consider only the *mpt* on a Lebesgue space.



Transformation extending all ergodic rotations
There is an ergodic system $\mathbf Z = (Z,m,R)$, such that $\mathbf Z$ admits every irrational rotation as a factor.


As mentioned in the comments, such a system $\mathbf Z$ cannot have $L^2(Z,m)$ separable, since the eigenspaces are mutually orthogonal, and there are uncountably many eigenspaces. However, $\mathbf Z$ has some nice properties: it is a rotation on a compact separable group $b\mathbb Z$, known as the *Bohr compactification* of the integers. There are several ways to construct $b\mathbb Z$, but the important properties of $b\mathbb Z$ are


* (B1) $b\mathbb Z$ is a compact abelian group (hence it has finite Haar measure $m$, which may be normalized to have $m(b\mathbb Z) = 1$).
* (B2) $\mathbb Z$ embeds densely in $b\mathbb Z$: there is a homomorphism $e:\mathbb Z \to b\mathbb Z$ having dense image in $b\mathbb Z$. Fixing such a homomorphism, we may consider $\mathbb Z$ to be a subgroup of $b\mathbb Z$.
* (B3) If $G$ is a compact abelian group and $\rho: \mathbb Z \to G$ is a homomoprhism, then $\rho$ extends to a continuous homomorphism $\rho: b\mathbb Z \to G$.


(See Rudin's *Fourier Analysis on Groups* for some exposition of the Bohr compactification.)


We may consider the measure preserving system $(Z,m,R)$, where $Z = b\mathbb Z$, $m=$ normalized Haar measure on $Z$, and $R z = z+1$. Then $(Z,m,R)$ is an ergodic group rotation, as the orbit of a point $z$ is $\{z+n : n\in \mathbb Z\}$, which is dense in $Z$, by (B2).


**Claim.** *Every* ergodic group rotation is a factor of $(Z,m,R)$.


*Proof.* Let $(G,m\_G,R\_\alpha)$ be an ergodic group rotation, so that $\{n\alpha : n \in \mathbb Z\}$ is dense in the compact abelian group $G$. The map $n\mapsto n\alpha$ is a homomorphism from $\mathbb Z$ to $G$ so Property (B3) provides a continuous homomorphism $\rho: b\mathbb Z\to G$ such that $\rho(n) = n\alpha$ for $n\in \mathbb Z.$ The continuity of $\rho$ then implies $\rho(b\mathbb Z) = G$, and it is easy to check that $\rho$ is actually a factor map. $\square$


It is natural to ask if $\mathbf Z$ is minimal among ergodic systems admitting every irrational rotation as a factor. The natural attempt to prove minimality would rely on the following two purported theorems, both of which are standard in the settings where $L^2(X,\mu)$ is separable, but I am not aware of their status in the non-separable setting.


**(purported) Theorem 1.** If $\mathbf X = (X,\mathcal B,\mu, T)$ is a measure preserving system and $\mathcal A$ is a $T$-invariant sub-sigma algebra of $\mathcal B$, then there is a factor $(Y,\mathcal D,\nu,S)$ of $\mathbf X$ with factor map $\phi: X\to Y$ such that


(i) $\phi^{-1}(D) \in \mathcal A$ (up to $\mu$-measure 0) for all $D\in \mathcal D$


(ii) For every $B\in \mathcal A$, there is $D\in \mathcal D$ such that $\mu(\phi^{-1}(D) \triangle B) = 0.$


**(purported) Theorem 2.** If $\mathbf Y = (Y,\nu,S)$ is an ergodic measure preserving system such that $L^2(Y,\nu)$ is spanned by the eigenfunctions of $\mathbf Y$, then $\mathbf Y$ is isomorphic to an ergodic group rotation.


Purported Theorem 2 is a generalization of the Halmos-von Neumann theorem to the non-separable setting.


