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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.
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 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)
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)
This article is rated Start-class on Wikipedia's
content assessment scale. It is of interest to the following WikiProjects: | ||||||||||||||||||||||||
|
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.
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 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)
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)