I am interested in the ergodic (invertible) transformations $T$ such that $T\times R\_\theta$ is ergodic where $R\_\theta$ is the rotation on $S^1$ with a given irrational angle $\theta$ (not all $R\_\theta$, only this one).


This includes all weakly mixing transformations, because of the two following results which can be found in Rudolph's book:



> 
> If $T$ is ergodic and has no factor isomorphic to an isometry on a
>  compact space, then $T \times S$ is ergodic for every ergodic
>  transformation $S$.
> 
> 
> 


and 



> 
> If $T$ is weakly mixing then it has no factor isomorphic to an
>  isometry on a compact space.
> 
> 
> 


There are other suitable $T$: every rotation $R\_\alpha$ fulfills the desideratum when $\alpha$ and $\theta$ are rationally independent. 


Can we expect a characterization of the suitable $T$ ? Or say something more ? Rotations $R\_\alpha$ are not suitable when $\alpha$ and $\theta$ are not rationally independent. Are there other transformations which are not suitable ?



Transformations whose product with a given rotation are ergodic
$T\times R\_\theta$ is ergodic if and only if $T$ is ergodic and $e^{2\pi im\theta}$ is not an eigenvalue of $T$ for any $m\in \mathbb Z\setminus\{0\}$.


For an idea of the proof, let $T\colon X\to X$. Then $L^2(X)$ can be decomposed as an orthogonal direct sum $V\_c\oplus V\_d$, the functions with continuous spectrum and the functions with discrete spectrum. $V\_d$ is spanned by the eigenfunctions of $T$. Similarly $L^2(S^1)$ has the standard Fourier basis, with $e\_k(x)=e^{2\pi ikx}$ being an eigenfunction of $R\_\theta$ with eigenvalue $e^{2\pi ik\theta}$. 


Now suppose there's an invariant function $F(x,t)$ for the product transformation. 
It can be decomposed as $\sum\_k (g\_k+h\_k)\otimes e\_k$, where $g\_k\in V\_c$ and $h\_k\in V\_d$. 
Since $F$ is invariant, we have $\frac 1N\sum\_{i=0}^{N-1}F\circ(T\times R)^i=F$. 


On the other hand, $\frac 1N\sum\_{i=0}^{N-1}(g\_k\otimes e\_k)\circ (T\times R)^i\to 0$ in $L^2$ by some spectral theory. Also $h\_k\otimes e\_k$ is a sum of eigenfunctions of $T\times R$. Any eigenfunction with eigenvalue other than 1 disappears under averaging. 


Hence if $F$ is invariant, it must be of the form $\sum\_k h\_k\otimes e\_k$ where $h\_k\otimes e\_k$ is an invariant function of $T\times R$, which implies $h\_k$ must satisfy $h\_k(Tx)=e^{-2\pi ik\theta}h\_k(x)$. Since by assumption $e^{2\pi i k\theta}$ is not an eigenvalue of $T$ for any $k$, all of the $h\_k$ must be 0
other than the constant term. So we deduce any invariant function for the product map is constant.


