Let $G$ be a discrete countable infinite group appearing on a compact metric house $X$ by way of homeomorphisms preserving a likelihood measure $mu$.
A operate $lambdacolon Gto mathbb C$ is an eigenvalue of the motion of $G$ if there exists a operate $fin L^2(X,mu)$ such that for each $gin G$ one has $lambda(g)cdot f=fcirc g$.
On this paper: Ergodicity of the Cartesian product by E. Flytzanis, Trans. Amer. Math. Soc. 186 (1973), 171-176 (freely accessible hyperlink at AMS web site), there are two outcomes concering the product of two dynamical programs given by $mathbf Z$-action:
- a enough situation for the ergodicity of the product,
- an outline of the spectrum of their cartesian product.
Additionally it is written there that the above outcomes maintain additionally for $G$-action, however it isn’t specified what does it precisely imply. Am I proper that:
- A product of $(X,G)$ and $(Y,G)$ is ergodic if the operate
continually equal to 1 is the one frequent eigenvalue for $(X,G)$ and
- The set of eigenvalues of $Xtimes Y$ equals the set of all capabilities
of the shape $fcdot g$ (pointwise multiplication), the place $f$ is an
eigenvalue for $X$ and $g$ is an eigenvalue for $Y$?
I want additionally two different properties, however I couldn’t discover the suitable references for them (possibly I’m mistaken that they’re true?):
- The set of eigenvalues of an element of a dynamical system $(X,G, mu)$
is contained within the set of eigenvalues of $(X,G,mu)$?
- Each eigenvalue of an ergodic dynamical system is straightforward (beware: I
don’t assume that $G$ is abelian).