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Letters in Mathematical Physics

, Volume 74, Issue 2, pp 111–133 | Cite as

Weak Mixing in Interval Exchange Transformations of Periodic Type

  • YA. G. Sinai
  • C. Ulcigrai
Article

Abstract

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.

Keywords

interval exchange transformations weak mixing spectral measures 

Mathematics Subject Classifications (2000)

37A05 37E05 37A30 

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References

  1. Avila, A., Forni, G.: Weak mixing for interval exchange transformations and translations flows. Ann. Math. (to appear). Preprint arXiv:math.DS/0406326Google Scholar
  2. 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
  3. Cornfeld I.P., Fomin S.V., Sinai Ya.G. (1980). Ergodic theory, Springer, Berlin Heidelberg New YorkzbMATHGoogle Scholar
  4. 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
  5. Katok A.B. (1980). Interval exchange transformations and some special flows are not mixing. Israel J. Math. 35(4): 301–310zbMATHCrossRefMathSciNetGoogle Scholar
  6. Keane M. (1975). Interval exchange trasformations. Mathematische Zeitschrift 141, 25–31zbMATHCrossRefMathSciNetGoogle Scholar
  7. Masur H. (1982). Interval exchange transformations and measured foliations. Ann. Math. 115, 169–200CrossRefMathSciNetGoogle Scholar
  8. 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
  9. Rauzy G. (1979). Échanges d’Intervalles et trasformations induites. Acta Arithmetica XXXIV, 315–328Google Scholar
  10. 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
  11. Veech W.A. (1982). Gauss measures for transformations on the space of interval exchange maps. Ann. Math. 115, 201–242CrossRefMathSciNetGoogle Scholar
  12. Veech W.A. (1984). The metric theory of interval exchange transformations I. Generic spectral properties. Am J Math. 107(6): 1331–1359MathSciNetGoogle Scholar
  13. Zorich A. (1996). Finite Gauss measure on the space of interval exchange transformation. Lyapunov exponents. Ann. Inst. Fourier, Grenoble. 46, 325–370zbMATHMathSciNetGoogle Scholar
  14. Zorich A. (1997). Deviation for interval exchange transformations. Ergodic Theory Dyn. Syst. 17, 1477–1499zbMATHCrossRefMathSciNetGoogle Scholar

Copyright information

© Springer 2005

Authors and Affiliations

  1. 1.Mathematics DepartmentPrinceton UniversityPrincetonUSA
  2. 2.Landau Institute of Theoretical PhysicsRussian Academy of SciencesMoscowRussia

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