Properties
Any interval exchange transformation is a bijection of to itself which preserves Lebesgue measure and is locally a translation. 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, a deep theorem due independently to W.Veech and to H.Masur 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 .
Read more about this topic: Interval Exchange Transformation
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