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I would like to compare minimal homeomorphisms and distal homeomorphisms on a compact metric space.

$f$ is said to be minimal if $\overline{Orb(x)}=X$ for all $x \in X$.

And $f$ is said to be distal if for $x \neq y$ then $\inf_{n \in \mathbb{z}} \{ d(f^n(x),f^n(y)) \} >0$.

It is easy to see that distal homeomorphism does not imply minimal homeomorphism.

For example: Identity map on an interval.

But, I have no idea about the converse?

Could you please help me find a counter example or proof for it?

user69833
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1 Answers1

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If I understand well, any minimal and mixing system will suffice. A standard example (in continuous time) is the horocycle flow one constant negative curvature surface. I know there are also more recent discrete-time examples, but I am not familiar with them.

demitau
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