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I'm reading a paper (Bullett and Sentenac, "Ordered orbits of the shift...", Ergodic Theory and Dynamical Systems), and have found that a proposition (Proposition 1) is (1) slightly incorrect (I have found a counterexample to the proposition that is stated); and (2) implicitly uses a lemma that is not stated in the paper.

The following is the statement of the implicit lemma:

Lemma Let $A$ be a subset of a circle $C$ such that

  • $A$ is not contained in any closed semi-circle of $C$; and
  • $A$ does not consist exactly of two pairs of diametrically opposite points.
Then there exist three points $a,b,c$ in $A$ that do not lie in any closed semi-circle of $C$ (i.e. there is a 3-point witness to the fact that $A$ does not lie in a semi-circle).

I have a proof, but it's less clean than I would like (with a couple of cases). I feel this should either be a consequence of a well known theorem, or there should be a very simple clean proof. Any suggestions?

anthonyquas
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  • I suppose you refer to https://www.cambridge.org/core/journals/mathematical-proceedings-of-the-cambridge-philosophical-society/article/ordered-orbits-of-the-shift-square-roots-and-the-devils-staircase/FCF6897070095CB61A71F2858D76772E The author did state just above Proposition 1 that A has a property $\sigma(A)=A$ which doesn’t apply for your counterexample. And anyway they didn’t consider cases when an opposite pairs exist because degenerate image is still considered ‘order-preserving’. For the lemma if your ‘a couple cases’ mean every point has its opposite or not I have nothing to add. – Ahmbak Mar 10 '18 at 14:38
  • Doesn’t $\sigma(A)={0,1/2}$in your case? – Ahmbak Mar 10 '18 at 15:03
  • Hmmm. True. Sorry I was thinking $\sigma(A)\subset A$. Of course I don’t think it’s clear in the proof in the paper how $\sigma(A)=A$ is used rather than $\sigma(A)\subset A$. – anthonyquas Mar 10 '18 at 15:06
  • Yeah. It seems they forgot or thought it was obvious that it would lead to contradiction though I’m not sure. Still the statement is correct. – Ahmbak Mar 10 '18 at 15:46

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