Interval exchange transformation

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. They arise naturally in the study of polygonal billiards and in area-preserving flows.

Graph of interval exchange transformation (in black) with and . In blue, the orbit generated starting from .

Formal definition

edit

Let   and let   be a permutation on  . Consider a vector   of positive real numbers (the widths of the subintervals), satisfying

 

Define a map   called the interval exchange transformation associated with the pair   as follows. For   let

 

Then for  , define

 

if   lies in the subinterval  . Thus   acts on each subinterval of the form   by a translation, and it rearranges these subintervals so that the subinterval at position   is moved to position  .

Properties

edit

Any interval exchange transformation   is a bijection of   to itself that preserves the Lebesgue measure. It is continuous except at a finite number of points.

The inverse of the interval exchange transformation   is again an interval exchange transformation. In fact, it is the transformation   where   for all  .

If   and   (in cycle notation), and if we join up the ends of the interval to make a circle, then   is just a circle rotation. The Weyl equidistribution theorem then asserts that if the length   is irrational, then   is uniquely ergodic. Roughly speaking, this means that the orbits of points of   are uniformly evenly distributed. On the other hand, if   is rational then each point of the interval is periodic, and the period is the denominator of   (written in lowest terms).

If  , and provided   satisfies certain non-degeneracy conditions (namely there is no integer   such that  ), a deep theorem which was a conjecture of M.Keane and due independently to William A. Veech[2] and to Howard Masur[3] asserts that for almost all choices of   in the unit simplex   the interval exchange transformation   is again uniquely ergodic. However, for   there also exist choices of   so that   is ergodic but not uniquely ergodic. Even in these cases, the number of ergodic invariant measures of   is finite, and is at most  .

Interval maps have a topological entropy of zero.[4]

Odometers

edit
 
Dyadic odometer  
 
Dyadic odometer iterated twice; that is  
 
Dyadic odometer thrice iterated; that is  
 
Dyadic odometer iterated four times; that is  

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

 

defined on the Cantor space   The standard mapping from Cantor space into the unit interval is given by

 

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

edit

Two and higher-dimensional generalizations include polygon exchanges, polyhedral exchanges and piecewise isometries.[5]

See also

edit

Notes

edit
  1. ^ Keane, Michael (1975), "Interval exchange transformations", Mathematische Zeitschrift, 141: 25–31, doi:10.1007/BF01236981, MR 0357739.
  2. ^ Veech, William A. (1982), "Gauss measures for transformations on the space of interval exchange maps", Annals of Mathematics, Second Series, 115 (1): 201–242, doi:10.2307/1971391, MR 0644019.
  3. ^ Masur, Howard (1982), "Interval exchange transformations and measured foliations", Annals of Mathematics, Second Series, 115 (1): 169–200, doi:10.2307/1971341, MR 0644018.
  4. ^ Matthew Nicol and Karl Petersen, (2009) "Ergodic Theory: Basic Examples and Constructions", Encyclopedia of Complexity and Systems Science, Springer https://doi.org/10.1007/978-0-387-30440-3_177
  5. ^ Piecewise isometries – an emerging area of dynamical systems, Arek Goetz

References

edit