Interval exchange transformation

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Graph of interval exchange transformation (in black) with \lambda = (1/15,2/15,3/15,4/15,5/15) and \pi=(3,5,2,4,1). In blue, the orbit generated starting from 1/2.

In mathematics, an interval exchange transformation[1] is a kind of dynamical system that generalises 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.

Formal definition[edit]

Let n > 0 and let \pi be a permutation on 1, \dots, n. Consider a vector \lambda = (\lambda_1, \dots, \lambda_n) of positive real numbers (the widths of the subintervals), satisfying

\sum_{i=1}^n \lambda_i = 1.

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

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)}.

Then for x \in [0,1], define


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

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

Properties[edit]

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

The inverse of the interval exchange transformation T_{\pi,\lambda} is again an interval exchange transformation. In fact, it is the transformation T_{\pi^{-1}, \lambda'} where \lambda'_i = \lambda_{\pi^{-1}(i)} for all 1 \leq i \leq n.

If n=2 and \pi = (12) (in cycle notation), and if we join up the ends of the interval to make a circle, then T_{\pi,\lambda} is just a circle rotation. The Weyl equidistribution theorem then asserts that if the length \lambda_1 is irrational, then T_{\pi,\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 \lambda_1 is rational then each point of the interval is periodic, and the period is the denominator of \lambda_1 (written in lowest terms).

If n>2, and provided \pi satisfies certain non-degeneracy conditions (namely there is no integer 0 < k < n such that \pi(\{1,\dots,k\}) = \{1,\dots,k\}), a deep theorem which was a conjecture of M.Keane and due independently to W.Veech [2] and to H.Masur [3] asserts that for almost all choices of \lambda in the unit simplex \{(t_1, \dots, t_n) : \sum t_i = 1\} the interval exchange transformation T_{\pi,\lambda} is again uniquely ergodic. However, for n \geq 4 there also exist choices of (\pi,\lambda) so that T_{\pi,\lambda} is ergodic but not uniquely ergodic. Even in these cases, the number of ergodic invariant measures of T_{\pi,\lambda} is finite, and is at most n.

Generalizations[edit]

Two and higher-dimensional generalizations include polygon exchanges, polyhedral exchanges and Piecewise isometries[4]

Notes[edit]

  1. ^ Michael Keane, Interval exchange transformations, Mathematische Zeitschrift 141, 25 (1975), http://www.springerlink.com/content/q10w48161l15gg18/
  2. ^ William A. Veech, Gauss measures for transformations on the space of interval exchange maps, Annals of Mathematics 115 (1982), http://www.jstor.org/stable/1971391
  3. ^ Howard Masur, Interval Exchange Transformations and Measured Foliations, Annals of Mathematics 115 (1982) http://www.jstor.org/stable/1971341
  4. ^ Piecewise isometries - an emerging area of dynamical systems, Arek Goetz

References[edit]