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" width=500 height=500)

Is there a precise example of a dynamical system whose minimality is not easy to proof but the symplicity of corresponding algebra is accessible and easy to investigation?

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

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?

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.3.8 of page 229 of this book.

Is there a precise example of a dynamical system whose minimality is not easy to proof but the symplicity of corresponding algebra is accessible and easy to investigation?

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

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" width=500 height=500)

Is there a precise example of a dynamical system whose minimality is not easy to proof but the symplicity of corresponding algebra is accessible and easy to investigation?

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

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 $C^*(X,\alpha)$. This algebra is simple if and only if the dynamical system $(X,\alpha)$ is a minimal dynamical system

Is there a precise example of a dynamical system whose minimality is not easy to proof but the symplicity of corresponding algebra is accessible and easy to investigation?

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

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.3.8 of page 229 of this book.

Is there a precise example of a dynamical system whose minimality is not easy to proof but the symplicity of corresponding algebra is accessible and easy to investigation?

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