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[[Image:Interval exchange.svg|thumb|right|Graph of interval exchange transformation (in black) with <math>\lambda = (1/15,2/15,3/15,4/15,5/15)</math> and <math>\pi=(3,5,2,4,1)</math>. In blue, the orbit generated starting from <math>1/2</math>.]]
In [[mathematics]], an '''interval exchange transformation''' <ref>Michael Keane, ''Interval exchange transformations'', Mathematische Zeitschrift 141, 25 (1975), ''http://www.springerlink.com/content/q10w48161l15gg18/'' (</ref> is a kind of [[dynamical system]] that generalises the idea of a [[circle rotation]]. The phase space consists of the [[unit interval]], and the transformation acts by cutting the interval into several subintervals, and then permuting these subintervals.▼
In [[mathematics]], an '''interval exchange transformation'''<ref>{{citation
| last = Keane | first = Michael
| doi = 10.1007/BF01236981
| journal = [[Mathematische Zeitschrift]]
| mr = 0357739
| pages = 25–31
| title = Interval exchange transformations
| volume = 141
▲
==Formal definition==
Let <math>n > 0</math> and let <math>\pi</math> be a [[permutation]] on <math>1, \dots, n</math>. Consider a [[Vector (geometric)|vector]] <math>\lambda = (\lambda_1, \dots, \lambda_n)</math> of positive real numbers (the widths of the subintervals), satisfying▼
▲Let <math>n > 0</math> and let <math>\pi</math> be a [[permutation]] on <math>1, \dots, n</math>. Consider a [[Vector (geometric)|vector]]
:<math>\sum_{i=1}^n \lambda_i = 1.</math>
Define a map <math>T_{\pi,\lambda}:[0,1]\rightarrow [0,1],</math> called the '''interval exchange transformation associated
:<math>a_i = \sum_{1 \leq j < i} \lambda_j \quad \text{and} \quad a'_i = \sum_{1 \leq j < \pi(i)} \lambda_{\pi^{-1}(j)}.</math>▼
▲called the '''interval exchange transformation associated to the pair <math>(\pi,\lambda)</math>''' as follows. For
▲:<math>a'_i = \sum_{1 \leq j < \pi(i)} \lambda_{\pi^{-1}(j)}.</math>
Then for <math>x \in [0,1]</math>, define
Line 31 ⟶ 25:
</math>
if <math>x</math> lies in the subinterval <math>[a_i,a_i+\lambda_i)</math>. Thus <math>T_{\pi,\lambda}</math> acts on each subinterval of the form <math>[a_i,a_i+\lambda_i)</math> by
==Properties==
Any interval exchange transformation <math>T_{\pi,\lambda}</math> is a [[bijection]] of <math>[0,1]</math> to itself that preserves the [[Lebesgue measure]]. It is continuous except at a finite number of points.
If <math>n=2</math> and <math>\pi = (12)</math> (in [[cycle notation]]), and if we join up the ends of the interval to make a circle, then <math>T_{\pi,\lambda}</math> is just a [[irrational rotation|circle rotation]]. The [[Weyl equidistribution theorem]] then asserts that if the length <math>\lambda_1</math> is [[irrational]], then <math>T_{\pi,\lambda}</math> is [[uniquely ergodic]].
If <math>n>2</math>, and provided <math>\pi</math> satisfies certain non-degeneracy conditions (namely there is no integer <math>0 < k < n</math> such that <math>\pi(\{1,\dots,k\}) = \{1,\dots,k\}</math>), a deep theorem which was a conjecture of M.Keane and due independently to [[William A. Veech]]<ref>{{citation
▲If <math>n=2</math> and <math>\pi = (12)</math> (in [[cycle notation]]), and if we join up the ends of the interval to make a circle, then <math>T_{\pi,\lambda}</math> is just a circle rotation. The [[Weyl equidistribution theorem]] then asserts that if the length <math>\lambda_1</math> is [[irrational]], then <math>T_{\pi,\lambda}</math> is [[uniquely ergodic]]. Roughly speaking, this means that the orbits of points of <math>[0,1]</math> are uniformly evenly distributed. On the other hand, if <math>\lambda_1</math> is rational then each point of the interval is [[Periodicity|periodic]], and the period is the denominator of <math>\lambda_1</math> (written in lowest terms).
| 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}}.</ref> and to [[Howard Masur]]<ref>{{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
Interval maps have a [[topological entropy]] of zero.<ref name="nicol">
▲If <math>n>2</math>, and provided <math>\pi</math> satisfies certain non-degeneracy conditions, a deep theorem due independently to W.Veech and to H.Masur asserts that for [[almost all]] choices of <math>\lambda</math> in the unit simplex <math>\{(t_1, \dots, t_n) : \sum t_i = 1\}</math> the interval exchange transformation <math>T_{\pi,\lambda}</math> is again [[uniquely ergodic]]. However, for <math>n \geq 4</math> there also exist choices of <math>(\pi,\lambda)</math> so that <math>T_{\pi,\lambda}</math> is [[ergodic]] but not [[uniquely ergodic]]. Even in these cases, the number of ergodic [[invariant]] [[measures]] of <math>T_{\pi,\lambda}</math> is finite, and is at most <math>n</math>.
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
</ref>
==
[[File:Dyadic odometer.svg|thumb|Dyadic odometer <math>T</math>]]
[[File:Dyadic odometer, twice iterated.svg|thumb|Dyadic odometer iterated twice; that is <math>T^2.</math>]]
[[File:Dyadic odometer thrice iterated.svg|thumb|Dyadic odometer thrice iterated; that is <math>T^3.</math>]]
[[File:Dyadic odometer iterated four times.svg|thumb|Dyadic odometer iterated four times; that is <math>T^4.</math>]]
The [[Markov odometer|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
:<math>T\left(1,\dots,1,0,b_{k+1},b_{k+2},\dots\right) = \left(0,\dots,0,1,b_{k+1},b_{k+2},\dots \right)</math>
defined on the [[Cantor space]] <math>\{0,1\}^\mathbb{N}.</math> The standard mapping from Cantor space into the [[unit interval]] is given by
:<math>(b_0,b_1,b_2,\cdots)\mapsto x=\sum_{n=0}^\infty b_n2^{-n-1}</math>
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.
== Higher dimensions==
Two and higher-dimensional generalizations include polygon exchanges, polyhedral exchanges and [[Piecewise isometry|
==See also==
* [[Markov odometer|Odometer]]
==Notes==
{{reflist}}▼
== References ==
* Artur Avila and Giovanni Forni, ''Weak mixing for interval exchange transformations and translation flows'', arXiv:math/0406326v1, ''
▲{{reflist}}
{{Chaos theory}}
▲* Artur Avila and Giovanni Forni, ''Weak mixing for interval exchange transformations and translation flows'', arXiv:math/0406326v1, ''http://arxiv.org/abs/math.DS/0406326''
[[Category:Chaotic maps]]
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