Let $G$ be a discrete countable infinite group acting on a compact metric space $X$ via homeomorphisms preserving a probability measure $\mu$. 


A function $\lambda\colon G\to \mathbb C$ is an *eigenvalue* of the action of $G$ if there exists a function $f\in L^2(X,\mu)$ such that for every $g\in G$ one has $\lambda(g)\cdot f=f\circ g$. 


In this paper: *Ergodicity of the Cartesian product* by E. Flytzanis, Trans. Amer. Math. Soc. 186 (1973), 171-176 ([freely available link at AMS site](https://www.ams.org/journals/tran/1973-186-00/S0002-9947-1973-0328021-6/)), there are two results concering the product of two dynamical systems given by $\mathbf Z$-action:


* a sufficient condition for the ergodicity of the product,
* a description of the spectrum of their cartesian product.


It is also written there that the above results hold also for $G$-action, but it is not specified what does it exactly mean. Am I right that:


* A product of $(X,G)$ and $(Y,G)$ is ergodic if the function
constantly equal to 1 is the only common eigenvalue for $(X,G)$ and
$(Y,G)$?
* The set of eigenvalues of $X\times Y$ equals the set of all functions
of the form $f\cdot g$ (pointwise multiplication), where $f$ is an
eigenvalue for $X$ and $g$ is an eigenvalue for $Y$?


I need also two other properties, but I could not find the appropriate references for them (maybe I am wrong that they are true?):


* The set of eigenvalues of a factor of a dynamical system $(X,G, \mu)$
is contained in the set of eigenvalues of $(X,G,\mu)$?
* Every eigenvalue of an ergodic dynamical system is simple (beware: I
do not assume that $G$ is abelian).



Properties of the spectrum of the Koopman representation
The answer to both question is negative.


Take $G=S\_3$, the symmetric group of the set $X=\{1,2,3\}$.
Then $L^2(X)$ is decomposed to the trivial representation and a another two dimensional irreducible representation.
As a $G$-space, $X\times X\simeq X \cup G$ where $X\subset X\times X$ is the diagonal and $G$ corresponds to the rest. 
This shows that $X\times X$ is not ergodic, providing a counter example to the first question.
It also follows that $L^2(X\times X)$ contains a sub-representation isomorphic to the regular representation $L^2(G)$ and in particular it contains the one dimensional sign representation. This provides a counter example to the second question.


