Question

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?

Answer 1

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.