Almost all interval exchange transformations with flips are nonergodic. (English) Zbl 0697.58027

Nogueira, Arnaldo

Ergodic Theory Dyn. Syst. 9, No. 3, 515-525 (1989).

The author first considers interval exchange transformations of the reals which are given by the successive lengths \(\lambda =(\lambda_ i)\) of the intervals and a permutation \(\sigma\) of those intervals. Generally such transformations have good ergodic properties. By results of Keane, Masur and Veech they are uniquely ergodic and minimal (i.e. have dense orbits for almost all points) for almost all such \(\lambda\). However this situation changes dramatically, if we allow those transformation to additionally flip some of their intervals. In this case the transformations have a periodic point with derivative equal to -1, and this property is an open property. This generalizes an observation by Keane, that any interval exchange map with exactly one reversed interval is periodic. Next the author studies the number of such flipped periodic points and shows, that the set of all exchanges of n intervals with n flipped periodic orbits has positive measure. The set of exchanges with more than i, \(i<n\) flipped periodic orbits can even be shown to be open.
As an interesting application and generalization the author discusses a billiard problem with flips in a rectangle, then in a polygon. In these examples similar properties hold.

Reviewer: C.H.Cap

MSC:

37B99	Topological dynamics
26A18	Iteration of real functions in one variable
28D05	Measure-preserving transformations
37C25	Fixed points and periodic points of dynamical systems; fixed-point index theory, local dynamics

Keywords:

interval transformations; ergodic; periodic point; billiard problem

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