# Does essentially minimal imply minimal?

Suppose X is compact and totally disconnected space, and that phi a homeomorphism of X.

We say a subset Z of X is phi-invariant if phi(Z) = Z. A phi-invariant set is minimal if it is closed, phi-invariant, nonempty and the smallest of all such sets. We say (X,phi) is minimal if X itself is a minimal set.

An orbit of x in X is the set {phi^n(x) : n an integer}

A system (X,phi) is minimal iff every orbit is dense.

Given (X,phi) as above, and any point y in X. The system is "essentially minimal" if one of the following equivalent conditions hold: 1) For all x in X, y in { phi^n(x) : n >= 0, n an integer }. 2) For all x in X, y in { phi^n(x) : n < 0, n an integer }. 3) X contains a unique minimal set Y, and y in Y.

If a system is minimal, then condition 3 is satisfied (setting Y := X), and is hence essentially minimal.

Does essential minimality imply minimality?

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I don't follow your definition of essential minimality. Does the definition depend on the point y? That is, can a system be essentially minimal for y but not for y'? Also, surely you want to take a closure of the sets in (1) and (2), otherwise you don't invoke the topology at all. –  Vaughn Climenhaga Aug 25 '10 at 2:03
(Also, your question will be easier to read if you utilise the website's ability to handle TeX.) –  Vaughn Climenhaga Aug 25 '10 at 2:03