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Perhaps the most important application of the Axiom of Choice within the theory of dynamical systems (meaning here, compact Hausdorff spaces with a self-map) yields, within every dynamical system, the existence of at least one non-empty minimal set (meaning a closed, invariant subset itself containing no proper closed, invariant subset). Since every point in a minimal set is almost periodic, this gives the existence of almost periodic points in every dynamical system.

Can anyone tell me please, how much of AC one gets back, over ZF, by taking the existence of minimal sets in dynamical systems as an axiom?

Are there interesting classes uncountable compact Hausdorff spaces where one has the existence of minimal sets already from ZF?

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In Katok and Hasselblatt /Introduction to the theory of dynamical systems/, the exercise 3.3.4 is to show the existence of a minimal set in a Hausdorff compact metric space without the axiom of choice, the idea being to use the Hausdorff metric on compact subsets. (The remark is credited to S. Simpson). –  Maxime Bourrigan Dec 15 '10 at 9:11
    
FYI, there's an axiom-of-choice tag in case you would like to add that to your list of tags. –  Jason Dec 15 '10 at 10:08
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This is somewhat of a non-answer to the second question (I do need a little bit of AC) so I apologize. But I thought it might be worthwhile for its generality.

The uncountable example I had in mind is $\beta \mathbb{N}$, the set of ultrafilters on the natural numbers, i.e., the Stone–Čech compactification of the discrete space $\mathbb{N}$.

  • $(\beta \mathbb{N},+)$ is a semigroup (with + extending the usual addition on $\mathbb{N}$ to $\beta \mathbb{N}$ in such a way that addition with a fixed right hand side is continuous)
  • The shift $s(p)= 1+ p$ makes $(\beta \mathbb{N},s)$ a dynamical system (because addition with natural numbers is left-continuous, too)
  • Its minimal systems are exactly the minimal left ideals of the semigroup
  • Its cardinality is $2^{2^{\aleph_0}}$ (i.e., the size of the power set of the reals)
  • The minimal left ideals are universal minimal systems for discrete time, so one minimal left ideal induces minimal systems everywhere

To prove all this, if I'm not mistaken, you require "only"

  • The ultrafilter lemma for (the power set of) $\mathbb{N}$ so that you actually get the space (the ultrafilter lemma is strictly weaker than AC)
  • An application Zorn's Lemma to find a minimal left ideal

So that's not "a lot" of AC to get one minimal system everywhere (which is why I thought it'd be worthwhile).

On the other hand, if you assume, e.g., ZF+AD you do not find any free ultrafilters on $\mathbb{N}$, so no $\beta \mathbb{N}$, and I have no idea what the dynamics look like then...

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How is an application of Zorn's Lemma less than "a lot" of AC? –  David Feldman Dec 16 '10 at 3:59
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Well, logically speaking you could have AC fail everywhere else and yet this one application holds. Making a wild guess I could try to turn it around and conjecture: you might not get a lot of AC out of the existence of minimal systems because all of it combined might be equivalent to only that one instance of Zorn's Lemma. –  Peter Krautzberger Dec 16 '10 at 18:59
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