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

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
Full Text: DOI
[1] DOI: 10.2307/1971341 · Zbl 0497.28012 · doi:10.2307/1971341
[2] Keane, Israel J. Math. 26 pp 188– (1977)
[3] Sataev, Izv. Acad. Sci. USSR 39 pp 860– (1970)
[4] DOI: 10.1007/BF01236981 · Zbl 0278.28010 · doi:10.1007/BF01236981
[5] Kerckhoff, Ergod. Th.& Dynam. Sys 5 pp 257– (1985)
[6] DOI: 10.2307/1971391 · Zbl 0486.28014 · doi:10.2307/1971391
[7] Veech, J. d’Analyse Math. 33 pp 222– (1978)
[8] DOI: 10.1016/0022-0396(87)90161-6 · Zbl 0643.58003 · doi:10.1016/0022-0396(87)90161-6
[9] DOI: 10.1016/0022-0396(78)90048-7 · Zbl 0413.58018 · doi:10.1016/0022-0396(78)90048-7
[10] Veech, Progress in Mathematics 1 pp 113– (1981)
[11] Rauzy, Acta Arith 34 pp 315– (1979)
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