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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
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(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

1 Answer 1

up vote 3 down vote accepted

As far as I understand the author tried (with missing closure in conditions (1) and (2), as it was already pointed out) to reproduce the standard definition of an essentially minimal dynamical system as one which has a unique minimal subset, see, for instance, Definition 1.2 from

MR1194074 (94f:46096) Herman, Richard H.(1-MD); Putnam, Ian F.(3-DLHS); Skau, Christian F.(N-TRND) Ordered Bratteli diagrams, dimension groups and topological dynamics. Internat. J. Math. 3 (1992), no. 6, 827--864.

There is a lot of examples of essentially minimal systems which are not minimal, see the Introduction to

MR1944409 (2003k:37020) Matui, Hiroki(J-CHIBES-MI) Topological orbit equivalence of locally compact Cantor minimal systems. (English summary) Ergodic Theory Dynam. Systems 22 (2002), no. 6, 1871--1903.

The simplest is just to take the shift on the integers and to extend it to the one-point compactification.

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Wow, thank you both. –  Daniel Mansfield Aug 26 '10 at 10:28

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