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), 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).