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Consider an interval exchange transformation that is, a bijective piecewise continuous map $[0,1] \rightarrow [0,1] $ whose restriction to its continuity intervals are translations. Assume that it is minimal and non uniquely ergodic.

Do the ergodic probability measures necessarily give the same weight to each of its continuity intervals?

I would naively expect the answer to this question to be negative, but since I have no example to test it

Thank you all!

Selim

Edit As Uri Bader pointed out in the comments, conjugating by a well chosen IET, any minimal non-uniquely ergodic IET provides a negative answer to the question, but the price to pay is to increase the number of continuity intervals involved. The 'good' question concerns then the IETs whose number of interval is minimal in its conjugacy class.

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  • $\begingroup$ When you say "do the measures give the same weight to each of its intervals", do you mean continuity intervals? (If not, any two distinct measures give different measure to some interval). $\endgroup$
    – Anthony Quas
    Commented Sep 16, 2016 at 16:42
  • $\begingroup$ Yes of course that's what I meant :) $\endgroup$
    – Selim G
    Commented Sep 16, 2016 at 16:47
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    $\begingroup$ Maybe you could edit the question to make this clear? $\endgroup$
    – Anthony Quas
    Commented Sep 16, 2016 at 19:25
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    $\begingroup$ It seems to me that by conjugating with an iet which has one break point I can always break a continuity interval into two in an arbitrary inner point (which I will move to 0), so the answer is negative, but this feels like cheatting... $\endgroup$
    – Uri Bader
    Commented Sep 20, 2016 at 12:10
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    $\begingroup$ Selim: you mean, up to conjugating by an arbitrary interval exchange transformation. $\endgroup$
    – YCor
    Commented Sep 20, 2016 at 12:49

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The space of invariant probability measures of a minimal interval exchange transformation $T$ on $d$ intervals $I_1, \dots, I_d$ is parametrized by a subsimplex $\mathcal{M}$ of the standard simplex $\Delta=\{(x_1,\dots,x_d)\in\mathbb{R}^d_+: \sum x_i=1\}$ in the sense that:

1) a point $(x_1,\dots, x_d)\in\mathcal{M}$ has the form $(x_1,\dots, x_d) = (\mu(I_1),\dots,\mu(I_d))$ for a $T$-invariant probability measure $\mu$.

2) the ergodic $T$-invariant probability measures correspond to the extremal points of $\mathcal{M}$.

This fact is explained in Section 4 of Ferenczi's notes (as pointed out by Uri Bader in the comments) and also Yoccoz's 'Pisa lecture notes' (https://www.college-de-france.fr/media/jean-christophe-yoccoz/UPL15305_PisaLecturesJCY2007.pdf). Very roughly speaking, $\mathcal{M}$ is obtained as a decreasing intersection of the successive images of the standard simplex $\Delta$ under the projective actions of the matrices prescribed by repeated iteration of the Rauzy-Veech algorithm starting with $T$.

In the case of a minimal non-uniquely ergodic interval exchange transformation $T$, the distinct ergodic invariant probability measures correspond to distinct points in $\Delta$ and, hence, it is always the case that some (actually, all but possibly one) ergodic probability measure does not give the same weight to all (continuity) intervals.

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    $\begingroup$ Thank you very much Matheus for taking the time to give this detailed answer! $\endgroup$
    – Selim G
    Commented Sep 25, 2016 at 14:06
  • $\begingroup$ You're welcome! $\endgroup$
    – Matheus
    Commented Sep 25, 2016 at 16:01

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