Skip to main content
SpringerLink
Log in

Quantitative weak mixing for interval exchange transformations

  • Published:
Geometric and Functional Analysis Aims and scope Submit manuscript

Abstract

We establish a dichotomy for the rate of the decay of the Cesàro averages of correlations of sufficiently regular functions for typical interval exchange transformations (IET) which are not rigid rotations (for which weak mixing had been previously established in the works of Katok–Stepin, Veech, and Avila–Forni). We show that the rate of decay is either logarithmic or polynomial, according to whether the IET is of rotation class (i.e., it can be obtained as the induced map of a rigid rotation) or not. In the latter case, we also establish that the spectral measures of Lipschitz functions have local dimension bounded away from zero (by a constant depending only on the number of intervals). In our approach, upper bounds are obtained through estimates of twisted Birkhoff sums of Lipschitz functions, while the logarithmic lower bounds are based on the slow deviation of ergodic averages that govern the relation between rigid rotations and their induced maps.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1

Similar content being viewed by others

References

  1. A. Avila and V. Delecroix. Weak mixing directions in non-arithmetic Veech surfaces, J. Amer. Math. Soc. 29.4 (2016), 1167–1208. https://doi.org/10.1090/jams/856

    Article  MathSciNet  MATH  Google Scholar 

  2. A. Avila and G. Forni. Weak mixing for interval exchange transformations and translation flows. Ann. Math. (2) 165.2 (2007), 637–664. https://doi.org/10.4007/annals.2007.165.637

  3. J.S. Athreya and G. Forni. Deviation of ergodic averages for rational polygonal billiards, Duke Math. J. 144.2 (2008), 285–319. https://doi.org/10.1215/00127094-2008-037

    Article  MathSciNet  MATH  Google Scholar 

  4. A. Avila, S. Gouëzel, and J.-C. Yoccoz. Exponential mixing for the Teichmüller flow, Publ. Math. Inst. Hautes Études Sci. 104 (2006), 143–211. https://doi.org/10.1007/s10240-006-0001-5.

    Article  MATH  Google Scholar 

  5. A. Avila and M. Leguil. Weak mixing properties of interval exchange transformations & translation flows, Bull. Soc. Math. France 146.2 (2018), 391–426. https://doi.org/10.24033/bsmf.2761.

  6. H. Al-Saqban et al. Exceptional directions for the Teichmüller geodesic flow and Hausdorff dimension, J. Eur. Math. Soc. 23.5 (2021), 1423–1476. https://doi.org/10.4171/JEMS/1037.

    Article  MathSciNet  MATH  Google Scholar 

  7. V.I. Arnol’d. Small denominators and problems of stability of motion in classical and celestial mechanics, Russian Mathematical Surveys 18.6 (1963), 85–191.

    Article  MathSciNet  MATH  Google Scholar 

  8. A. Avila and M. Viana. Simplicity of Lyapunov spectra: a sufficient criterion, Port. Math. (N.S.) 64.3 (2007), 311–376. https://doi.org/10.4171/PM/1789.

  9. A.I. Bufetov and B. Solomyak. On the modulus of continuity for spectral measures in substitution dynamics, Advances in Mathematics 260 (2014), 84–129.

    Article  MathSciNet  MATH  Google Scholar 

  10. A.I. Bufetov and B. Solomyak. The Hölder property for the spectrum of translation flows in genus two, Israel J. Math. 223.1 (2018), 205–259. https://doi.org/10.1007/s11856-017-1614-8.

    Article  MathSciNet  MATH  Google Scholar 

  11. A.I. Bufetov and B. Solomyak. Hölder regularity for the spectrum of translation flows, Journal de l’École polytechnique -Mathématiques 8 (2021), 279–310. https://doi.org/10.5802/jep.146.

    Article  MathSciNet  MATH  Google Scholar 

  12. A.I. Bufetov, Y.G. Sinai, and C. Ulcigrai. A condition for Continuous Spectrum of an Interval Exchange Transformation, American Mathematical Society Translations 217 (2006), 23–35.

    MathSciNet  MATH  Google Scholar 

  13. A.I. Bufetov. Limit theorems for suspension flows over Vershik automorphisms, Russian Mathematical Surveys 68.5 (2013), 789–860. https://doi.org/10.1070/rm2013v068n05abeh004858.

    Article  MathSciNet  MATH  Google Scholar 

  14. A.I. Bufetov. Limit theorems for translation flows, Annals of Mathematics 179 (2014), 431–499.

    Article  MathSciNet  MATH  Google Scholar 

  15. J. Chaves and A. Nogueira. Spectral properties of interval exchange maps, Monatsh. Math. 134.2 (2001), 89–102. https://doi.org/10.1007/s006050170001.

    Article  MathSciNet  MATH  Google Scholar 

  16. S. Ferenczi, C. Holton, and L.Q. Zamboni. Structure of three-interval exchange transformations III: Ergodic and spectral properties, Journal d’Analyse Mathématique 93 (2004), 103–138.

    Article  MathSciNet  MATH  Google Scholar 

  17. N.P. Fogg et al. Substitutions in Dynamics, Arithmetics and Combinatorics. Lecture Notes in Mathematics. Springer Berlin Heidelberg, 2003.

    Google Scholar 

  18. G. Forni. Deviation of ergodic averages for area-preserving flows on surfaces of higher genus, Ann. of Math. (2) 155.1 (2002), 1–103. https://doi.org/10.2307/3062150.

  19. G. Forni. Twisted Translation Flows and Effective Weak Mixing, J. Eur. Math. Soc. 24.12 (2022), 4225–4276. https://doi.org/10.4171/JEMS/1186.

    Article  MathSciNet  MATH  Google Scholar 

  20. M. Herman. Sur la conjugaison différentiable des difféomorphismes du cercle à des rotations, Publications Mathématiques de l’IHÉS 49 (1979), 5–233.

    Article  MATH  Google Scholar 

  21. A. Katok. Interval exchange transformations and some special flows are not mixing, Israel J. Math. 35.4 (1980), 301–310. https://doi.org/10.1007/BF02760655.

    Article  MathSciNet  MATH  Google Scholar 

  22. M. Keane. Interval exchange transformations, Math. Z. 141 (1975), 25–31. https://doi.org/10.1007/BF01236981.

    Article  MathSciNet  MATH  Google Scholar 

  23. A.Ya. Khinchin. Continued fractions. The University of Chicago Press, Chicago, Ill.- London, 1964, pp. xi+95.

  24. A.B. Katok and A.M. Stepin. Approximations in ergodic theory, Uspehi Mat. Nauk 22.5 (1967), 81–106.

    MathSciNet  MATH  Google Scholar 

  25. A. Katok and J.-P. Thouvenot. Chapter 11 - Spectral Properties and Combinatorial Constructions in Ergodic Theory. In: Handbook of Dynamical Systems. Ed. by B. Hasselblatt and A. Katok. Vol. 1. Handbook of Dynamical Systems. Elsevier Science, 2006, pp. 649–743. https://doi.org/10.1016/S1874-575X(06)80036-6.

  26. J. Marshall-Maldonado. Modulus of continuity for spectral measures of suspension flows over Salem type substitutions. arXiv e-prints, arXiv:2009.13607 (2020), pp. 1–19. arXiv:1908.09347 [math.DS].

  27. H. Masur. Interval Exchange Transformations and Measured Foliations, Annals of Mathematics 115.1 (1982), 169–200.

    Article  MathSciNet  MATH  Google Scholar 

  28. A. Nogueira and D. Rudolph. Topological weak-mixing of interval exchange maps, Ergodic Theory Dynam. Systems 17.5 (1997), 1183–1209. https://doi.org/10.1017/S0143385797086276.

    Article  MathSciNet  MATH  Google Scholar 

  29. V. Iustinovich Oseledec. A multiplicative ergodic theorem. Liapunov characteristic number for dynamical systems, Trans. Moscow Math. Soc. 19 (1968), 197–231.

  30. W. Parry. Topics in Ergodic Theory. Cambridge Tracts in Mathematics. Cambridge University Press, 2004.

  31. K.E. Petersen. Ergodic Theory. Cambridge Studies in Advanced Mathematics. Cambridge University Press, 1983. https://doi.org/10.1017/CBO9780511608728.

  32. G. Rauzy. Echanges d’intervalles et transformations induites, Acta Arithmetica 34.4 (1979), 315–328.

    Article  MathSciNet  MATH  Google Scholar 

  33. Y.G. Sinai and C. Ulcigrai. Weak Mixing in Interval Exchange Transformations of Periodic Type, Letters in Mathematical Physics 74 (2005), 111–133.

    Article  MathSciNet  MATH  Google Scholar 

  34. W.A. Veech. Gauss Measures for Transformations on the Space of Interval Exchange Maps, Annals of Mathematics 115.2 (1982), 201–242.

    Article  MathSciNet  MATH  Google Scholar 

  35. W.A. Veech. The Metric Theory of Interval Exchange Transformations I. Generic Spectral Properties, American Journal of Mathematics 106.6 (1984), 1331–1359.

    MathSciNet  MATH  Google Scholar 

  36. W.A. Veech. The Teichmuller Geodesic Flow, Annals of Mathematics 124.3 (1986), pp. 441–530.

    Article  MathSciNet  MATH  Google Scholar 

  37. M. Viana. Dynamics of interval exchange transformations and Teichmüller flows, Available at https://www.mat.univie.ac.at/ bruin/ietf.pdf (2008).

  38. J.-C. Yoccoz. Interval exchange maps and translation surfaces, Homogeneous flows, moduli spaces and arithmetic (Clay Mathematics Proceedings) 10 (2010), 1–69.

  39. E. Zehnder. Generalized implicit function theorems with applications to some small divisor problems, I, Communications on Pure and Applied Mathematics 28.1 (1975), 91–140. https://doi.org/10.1002/cpa.3160280104.

    Article  MathSciNet  MATH  Google Scholar 

  40. A. Zorich. Finite Gauss measure on the space of interval exchange transformations. Lyapunov exponents. Annales de l’Institut Fourier 46.2 (1996), 325–370. https://doi.org/10.5802/aif.1517.

    Article  MathSciNet  MATH  Google Scholar 

  41. A. Zorich. Deviation for interval exchange transformations, Ergodic Theory and Dynamical Systems 17.6 (1997), 1477–1499. https://doi.org/10.1017/S0143385797086215.

    Article  MathSciNet  MATH  Google Scholar 

Download references

Acknowledgements

During work on this project, Artur Avila and Pedram Safaee were both supported by the SNF grant 188535. Giovanni Forni was supported by the NSF grant DMS 2154208.

Author information

Authors and Affiliations

  1. Institut für Mathematik Universität Zürich, Winterthurerstrasse 190, 8057, Zurich, Switzerland

    Artur Avila & Pedram Safaee

  2. IMPA, Estrada Dona Castorina, 110, Rio de Janeiro, 22460-320, Brazil

    Artur Avila

  3. Department of Mathematics, University of Maryland, College Park, MD, 20742, USA

    Giovanni Forni

Authors
  1. Artur Avila
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Giovanni Forni
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Pedram Safaee
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Giovanni Forni.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Avila, A., Forni, G. & Safaee, P. Quantitative weak mixing for interval exchange transformations. Geom. Funct. Anal. 33, 1–56 (2023). https://doi.org/10.1007/s00039-023-00625-y

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00039-023-00625-y

Search

Navigation