Talk:Interval exchange transformation

Sources:
Cornfeld Fomin and Sinai's book gives a general construction of IET with n ergodic measures (I think there construction requires 2n+1 intervals.) Michael Keane's Non-ergodic interval exchange transformations, gives a non-uniquelly ergodic minimal 4 IET. (Keynes and Newton gave a non-uniquelly ergodic 5 IET based on an older example of W. Veech) Veech's proof that a.e. IET w/ irreducible permutation is uniquelly ergodic is in The Metric Theory of interval exchange transformations Masur's is in Interval exchange transformations and measured foliation.


 * Veech proof (1982) also works also for measured foliations as ergodicity is invariant under time change. Odiralgnirt

The bound for ergodic measure of an n interval IET is in Cornfeld Fomin and Sinai. For minimal IET's [n/2] is a bound as shown by Veech in Interval exchange transformations. Anatole Katok I think also have showed this.


 * The actual bound for interval exchanges is g where g is the genus of the associated measured foliation or translation surfaces. The original proof is by Katok ("Invariant measures of flows on orientable surfaces", Akad. Nauk SSSR 211, 1973) Odiralgnirt

Problems with visualization of the odometer
The section Odometers defines the odometer mapping on the Cantor space viewed as {0,1}ℕ. It also defines a mapping between this Cantor space and the unit interval [0,1]. And finally, the "visualization of the odometer" depicts a mapping from the unit interval to itself.

But this section never actually states how these three things are connected with each other. Although I have a guess, it would be good if someone knowledgeable on this subject filled in this missing information.

Also, the illustrations visualizing the odometer have a problem: Because they are all drawn so as to have a continuous graph, they make it appear that the odometer is a continuous map on the unit interval. As such, it is very far from being a bijection.

But it is in fact a bijection (actually a homeomorphism) of the Cantor set to itself. So it would be much better if the illustrations depicted the odometer mapping as a bijection. In other words, without connecting parts of the graph that should not be connected to each other. 2601:200:C000:1A0:EC27:E3E5:4AD9:D440 (talk) 23:54, 10 August 2021 (UTC)

Bad writing
The first sentence in the section Properties is the following:

"Any interval exchange transformation is a bijection of  to itself preserves the Lebesgue measure."

I hope someone knowledgeable about this subject can rewrite this so that it is readable. 2601:200:C000:1A0:BC00:5039:DB55:E9EC (talk) 21:55, 1 August 2022 (UTC)
 * Simplest fix seems to be to change "preserves" to "that preserves". —David Eppstein (talk) 22:40, 1 August 2022 (UTC)