interval exchange transformation

Let $n\; >\; 0$ and let $\backslash pi$ be a permutation on $1,\; \backslash dots,\; n$. Consider a vector $\backslash lambda\; =\; (\backslash lambda\_1,\; \backslash dots,\; \backslash lambda\_n)$ of positive real numbers (the widths of the subintervals), satisfying

:$\backslash sum\_\{i=1\}^n\; \backslash lambda\_i\; =\; 1.$

Define a map $T\_\{\backslash pi,\backslash lambda\}:[0,1]\backslash rightarrow\; [0,1],$ called the interval exchange transformation associated with the pair $(\backslash pi,\backslash lambda)$ as follows. For $1\; \backslash leq\; i\; \backslash leq\; n$ let

:

Then for $x\; \backslash in\; [0,1]$, define

:$$

T_{\pi,\lambda}(x) = x - a_i + a'_i

if $x$ lies in the subinterval $[a\_i,a\_i+\backslash lambda\_i)$. Thus $T\_\{\backslash pi,\backslash lambda\}$ acts on each subinterval of the form $[a\_i,a\_i+\backslash lambda\_i)$ by a translation, and it rearranges these subintervals so that the subinterval at position $i$ is moved to position $\backslash pi(i)$.

Any interval exchange transformation $T\_\{\backslash pi,\backslash lambda\}$ is a bijection of $[0,1]$ to itself that preserves the Lebesgue measure. It is continuous except at a finite number of points.

The inverse of the interval exchange transformation $T\_\{\backslash pi,\backslash lambda\}$ is again an interval exchange transformation. In fact, it is the transformation $T\_\{\backslash pi^\{-1\},\; \backslash lambda\text{'}\}$ where $\backslash lambda\text{'}\_i\; =\; \backslash lambda\_\{\backslash pi^\{-1\}(i)\}$ for all $1\; \backslash leq\; i\; \backslash leq\; n$.

If $n=2$ and $\backslash pi\; =\; (12)$ (in cycle notation), and if we join up the ends of the interval to make a circle, then $T\_\{\backslash pi,\backslash lambda\}$ is just a circle rotation. The Weyl equidistribution theorem then asserts that if the length $\backslash lambda\_1$ is irrational, then $T\_\{\backslash pi,\backslash lambda\}$ is uniquely ergodic. Roughly speaking, this means that the orbits of points of $[0,1]$ are uniformly evenly distributed. On the other hand, if $\backslash lambda\_1$ is rational then each point of the interval is periodic, and the period is the denominator of $\backslash lambda\_1$ (written in lowest terms).

If $n>2$, and provided $\backslash pi$ satisfies certain non-degeneracy conditions (namely there is no integer such that $\backslash pi(\backslash \{1,\backslash dots,k\backslash \})\; =\; \backslash \{1,\backslash dots,k\backslash \}$), a deep theorem which was a conjecture of M.Keane and due independently to William A. Veech{{citation

| last = Veech | first = William A. | authorlink = William A. Veech

| doi = 10.2307/1971391

| issue = 1

| journal = Annals of Mathematics

| mr = 644019

| pages = 201–242

| series = Second Series

| title = Gauss measures for transformations on the space of interval exchange maps

| volume = 115

| year = 1982}}. and to Howard Masur{{citation

| last = Masur | first = Howard

| doi = 10.2307/1971341

| issue = 1

| journal = Annals of Mathematics

| mr = 644018

| pages = 169–200

| series = Second Series

| title = Interval exchange transformations and measured foliations

| volume = 115

| year = 1982}}. asserts that for almost all choices of $\backslash lambda$ in the unit simplex $\backslash \{(t\_1,\; \backslash dots,\; t\_n)\; :\; \backslash sum\; t\_i\; =\; 1\backslash \}$ the interval exchange transformation $T\_\{\backslash pi,\backslash lambda\}$ is again uniquely ergodic. However, for $n\; \backslash geq\; 4$ there also exist choices of $(\backslash pi,\backslash lambda)$ so that $T\_\{\backslash pi,\backslash lambda\}$ is ergodic but not uniquely ergodic. Even in these cases, the number of ergodic invariant measures of $T\_\{\backslash pi,\backslash lambda\}$ is finite, and is at most $n$.

Interval maps have a topological entropy of zero.

Matthew Nicol and Karl Petersen, (2009) "[https://www.math.uh.edu/~nicol/pdffiles/petersen.pdf Ergodic Theory: Basic Examples and Constructions]",

Encyclopedia of Complexity and Systems Science, Springer https://doi.org/10.1007/978-0-387-30440-3_177

File:Dyadic odometer.svg

File:Dyadic odometer, twice iterated.svg

File:Dyadic odometer thrice iterated.svg

File:Dyadic odometer iterated four times.svg

The dyadic odometer can be understood as an interval exchange transformation of a countable number of intervals. The dyadic odometer is most easily written as the transformation

:$T\backslash left(1,\backslash dots,1,0,b\_\{k+1\},b\_\{k+2\},\backslash dots\backslash right)\; =\; \backslash left(0,\backslash dots,0,1,b\_\{k+1\},b\_\{k+2\},\backslash dots\; \backslash right)$

defined on the Cantor space $\backslash \{0,1\backslash \}^\backslash mathbb\{N\}.$ The standard mapping from Cantor space into the unit interval is given by

:$(b\_0,b\_1,b\_2,\backslash cdots)\backslash mapsto\; x=\backslash sum\_\{n=0\}^\backslash infty\; b\_n2^\{-n-1\}$

This mapping is a measure-preserving homomorphism from the Cantor set to the unit interval, in that it maps the standard Bernoulli measure on the Cantor set to the Lebesgue measure on the unit interval. A visualization of the odometer and its first three iterates appear on the right.

Two and higher-dimensional generalizations include polygon exchanges, polyhedral exchanges and piecewise isometries.[http://math.sfsu.edu/goetz/Research/graz/graz.pdf Piecewise isometries – an emerging area of dynamical systems], Arek Goetz

• Odometer

{{reflist}}

• Artur Avila and Giovanni Forni, Weak mixing for interval exchange transformations and translation flows, arXiv:math/0406326v1, https://arxiv.org/abs/math.DS/0406326

{{Chaos theory}}

Category:Chaotic maps