Weak Mixing in Interval Exchange Transformations of Periodic Type
- 117 Downloads
Interval exchange transformations (IETs) are piecewise isometries of the interval, obtained permuting a certain number of subintervals. We give a condition on IETs in the special subclass of IETs with periodic Rauzy-Veech cocycle which guarantees weak mixing, i.e. the continuity of the spectrum. The proof involves the study of the associated spectral measures. The condition can be checked explicitly by computing a certain Galois group of a field related to the Ravzy-Veech cocycle. Explicit examples of weakly mixing IETs are constructed in the Appendix.
Keywordsinterval exchange transformations weak mixing spectral measures
Mathematics Subject Classifications (2000)37A05 37E05 37A30
Unable to display preview. Download preview PDF.
- Avila, A., Forni, G.: Weak mixing for interval exchange transformations and translations flows. Ann. Math. (to appear). Preprint arXiv:math.DS/0406326Google Scholar
- Bufetov, A., Sinai, Y.G., Ulcigrai, C.: A condition for continuous spectrum of an interval exchange transformation. To appear in a volume of AMS dedicated to the 70th birthday of A.M. VershikGoogle Scholar
- Ferenczi S., Maudit C., Nogueira A. (1996). Substitution dynamical systems: algebraic characterization of eigenvalues. Ann. Scientifiques de l’É N.S. 4e Série, 29 (4), 519–533 (1996)Google Scholar
- Marmi, S., Moussa, P., Yoccoz, J.-C.: The cohomological equation for Roth type interval exchange maps. J. Am. Math. Soc. (to appear). http://www.ams.org/jams/0000-000-00/S0894-0347-05-00490-X/home.htmlGoogle Scholar
- Rauzy G. (1979). Échanges d’Intervalles et trasformations induites. Acta Arithmetica XXXIV, 315–328Google Scholar
- Veech, W.A.: Ergodic theory and dynamical systems, I, chapter. Projective Swiss cheeses and uniquely ergodic interval exchange transformations, pp. 113–193. College Park, Md., 1979–80. Birkhäuser (1981)Google Scholar