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Let $(X, d)$ be a compact metric space, let $T$ be a group of actions on $X$. Then $(X,T)$ is a topological dynamical system with transformation group $T$, and we denote it by $(X,T)$. We say points $x, y\in X$ are proximal when we have that $$\inf_{t\in T}\ d(tx, ty)=0.$$ A point $x\in X$ is called distal if it is only proximal to itself.

A topological dynamical system $(X,T)$ is called point-distal if there exists a point $x\in X$ such that $x$ is distal and the orbit of $x$ is dense in $X$.

Minimal systems which are not point-distal are not hard to be produced. But when $T$ is abelian, there does not seem to be a large store of examples, according to Veech's paper "point-distal flow"(MR0267560). In this paper, Veech gave a example of a type of minimal systems which are not point-distal: the horocycle flows.

My question is that: when $T=\mathbb{Z}$, except the horocycle flows, is there any other example of minimal dynamical system which is not point distal? I think such system exists, but I don't know how to construct one.

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  • $\begingroup$ If you do a rank one cutting and stacking construction, you can get something with this property if you add in more and more spacers all labelled with the same symbol. $\endgroup$ Mar 24, 2014 at 14:27
  • $\begingroup$ Your question is not very easy to spot, you should state it precisely in an isolated sentence and without ambiguity (e.g. avoiding "this kind"). $\endgroup$ Mar 24, 2014 at 18:48
  • $\begingroup$ @Benoît Kloeckner: Thank you for your advice. I have revised my question a little bit. $\endgroup$
    – Siming Tu
    Mar 30, 2014 at 1:58

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