Typical interval exchange transformation is either rotation or weakly mixing

Analysis 2 Minutes

You need to know: Permutation of \{1,2, \dots, d\}, notation {\mathbb R}_+^d for vectors in {\mathbb R}_d with non-negative components, notation f^k(x)=f(f(\dots f(x) \dots)) for k-fold function composition,  notation f^{-k}(A) for set \{x: f^k(x) \in A\}, notation a \equiv b (\text{mod } d) if a-b is divisible by d, Lebesgue measure, measurable sets, Lebesgue almost every.

Background: Let d \geq 2 be an integer. Permutation \pi of \{1,2, \dots, d\} is called irreducible if \pi (\{1,\dots,k\}) \neq \{1,\dots,k\} for all 1 \leq k<d. Given such \pi and \lambda = (\lambda_1, \dots, \lambda_d)\in {\mathbb R}_+^d, an interval exchange transformation f = f(\lambda, \pi) is a map f:I \to I, which divides I = \left[0,\sum\limits_{i=1}^d \lambda_i\right) into sub-intervals I_i = \left[\sum\limits_{j<i} \lambda_j,\sum\limits_{j \leq i} \lambda_j\right), \, i=1,2\dots,d and rearranges the I_i according to \pi (it maps every x \in I_i into x + \sum\limits_{\pi(j)<\pi(i)} \lambda_j - \sum\limits_{j<i} \lambda_j). f is called weakly mixing if for every pair of measurable sets A, B \subset I, \lim\limits_{n\to\infty} \frac{1}{n}\sum\limits_{k=1}^{n-1} \left|m(f^{-k}(A) \cap B) -m(A)m(B)\right| = 0, where m denotes the Lebesgue measure. Permutation \pi of \{1,2, \dots, d\} is called a rotation if \pi(i + 1) \equiv \pi(i) + 1 (\text{mod } d), for all i \in \{1,2, \dots, d\}.

The Theorem: On 16th June 2004, Artur Ávila and Giovanni Forni submitted to arxiv a paper in which they proved that for every irreducible permutation \pi of \{1,2, \dots, d\} which is not a rotation, and Lebesgue almost every \lambda\in {\mathbb R}_+^d, f(\lambda, \pi) is weakly mixing.

Short context: Interval exchange transformations (IETs in short) are basic examples of measure-preserving transformations f:I \to I (that is, such that m(f^{-1}(A))=m(A) for all measurable A \subset I), which are central objects of study in dynamical systems. f is called mixing if \lim\limits_{n\to\infty} m(f^{-n}(A) \cap B) = m(A)m(B) for all measurable A, B \subset I, and ergodic if f^{-1}(A)=A implies that m(A)=0 or m(A)=m(I). It is known that every mixing f is weakly mixing, and every weakly mixing f is ergodic. It was known that almost every IET is ergodic, and the Theorem proves a stronger result that almost every non-rotation IET is weakly mixing. Because, by 1980 theorem of Katok, IETs are not mixing, the weak mixing property is the strongest we could hope for.

Links: Free arxiv version of the original paper is here, journal version is here. See also Section 7.4 of this book for an accessible description of the Theorem.

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Published by Bogdan Grechuk

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2 thoughts on “Typical interval exchange transformation is either rotation or weakly mixing”

  1. Pingback: The product of typical interval exchange transformations is uniquely ergodic – Theorems of the 21st century
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