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Let $X$ be a compact Hausdorff space and $\alpha$ be a homeomorphism of $X$.

So we have a natural action of $\mathbb{Z}$ on $C(X)$ which generates the cross product algebra $C^*(X,\alpha)$. [It is well known that this algebra is simple if and only if the dynamical system $(X,\alpha)$ is a minimal dynamical system, see Corollary VIII.3.9 of page 229 of the book $C^*$ Algebras by examples by Kenneth R. Davidson, "https://books.google.com/books?id=0TXteNfrzvcC&lpg=PA223&vq=minimal%20dynamical%20system&pg=PA229&output=embed")

Is there a precise example of a dynamical system whose minimality is not easy to prove but the simplicity of corresponding algebra is accessible and easy to investigate?

Is there an example of a dynamical system with a finite number of minimal components such that the number of minimal components can be precisely observed in the corresponding cross product algebra?

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  • $\begingroup$ Regarding the second question, when there are no periodic points, there is a one-to-one correspondence between ideals in the crossed product algebra and closed invariant subsets of $X$, so the number of mininal components is equal to the number of maximal ideals in the crossed product. $\endgroup$
    – Ruy
    Commented Aug 12, 2019 at 21:34

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