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I am looking for an example of a dynamical system $(X,T)$ such that:

  • $X$ is a connected topological space,
  • $(X,T)$ is minimal and weakly mixing.

Does there exists one?

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There are examples of minimal real analytics torus diffeomorphisms which are not just weak mixing, but strong mixing.

In fact, Bassam Fayad has constructed some mixing reparametrizations of minimal linear flows on tori in this paper.

To get a diffeomorphism with those properties from such a flow, one can show that given any minimal flow, there is a generic set $\mathcal{R}\subset\mathbb{R}$ such that the time-$t$ map is indeed minimal for every $t\in\mathcal{R}$, and such map must be mixing.

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