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.
of positive real numbers (the widths of the subintervals), satisfying Define a map
called the interval exchange transformation associated with the pair
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
Any interval exchange transformation
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
(in cycle notation), and if we join up the ends of the interval to make a circle, then
The Weyl equidistribution theorem then asserts that if the length
Roughly speaking, this means that the orbits of points of
are uniformly evenly distributed.
is rational then each point of the interval is periodic, and the period is the denominator of
(written in lowest terms).
satisfies certain non-degeneracy conditions (namely there is no integer
), 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
the interval exchange transformation
Even in these cases, the number of ergodic invariant measures of
Interval maps have a topological entropy of zero.
[4] 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.
Two and higher-dimensional generalizations include polygon exchanges, polyhedral exchanges and piecewise isometries.
