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Quantitative shrinking target properties for rotations and interval exchanges

  • Jon Chaika
  • David ConstantineEmail author
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Abstract

This paper presents quantitative shrinking target results for rotations and interval exchange transformations. To do this a quantitative version of a unique ergodicity criterion of Boshernitzan is established.

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References

  1. [1]
    J. S. Athreya and G. Forni, Deviation of ergodic averages for rational polygonal billiards, Duke Mathematical Journal 144 (2008), 285–319.MathSciNetCrossRefzbMATHGoogle Scholar
  2. [2]
    M. Boshernitzan, Rank two interval exchange transformations, Ergodic Theory and Dynamical Systems 8 (1988), 379–394.MathSciNetCrossRefzbMATHGoogle Scholar
  3. [3]
    M. Boshernitzan, A condition for unique ergodicity of minimal symbolic flows, Ergodic Theory and Dynamical Systems 12 (1992), 425–428.MathSciNetCrossRefzbMATHGoogle Scholar
  4. [4]
    M. Boshernitzan and J. Chaika, Quantitative proximality and connectedness, Inventiones Mathematicae 192 (2013), 375–412.MathSciNetCrossRefzbMATHGoogle Scholar
  5. [5]
    N. Chernov and D. Kleinbock, Dynamical Borel–Cantelli lemmas for Gibbs measures, Israel Journal of Mathematics 122 (2001), 1–27.MathSciNetCrossRefzbMATHGoogle Scholar
  6. [6]
    D. Dolgopyat, Limit theorems for partially hyperbolic systems, Transactions of the American Mathematical Society 356 (2004), 1637–1689.MathSciNetCrossRefzbMATHGoogle Scholar
  7. [7]
    G. Forni, Deviation of ergodic averages for area-preserving flows on surfaces of higher genus, Annals of Mathematics 155 (2002), 1–103.MathSciNetCrossRefzbMATHGoogle Scholar
  8. [8]
    S. Galatolo, Hitting time and dimension in axiom A systems, generic interval exchanges and an application to Birkoff sums, Journal of Statistical Physics 123 (2006), 111–124.MathSciNetCrossRefzbMATHGoogle Scholar
  9. [9]
    A. Ya. Khinchin, Continued Fractions, Dover, Mineola, NY, 1997.zbMATHGoogle Scholar
  10. [10]
    D. H. Kim and S. Marmi, The recurrence time for interval exchange maps, Nonlinearity 21 (2008), 2201–2210.MathSciNetCrossRefzbMATHGoogle Scholar
  11. [11]
    U. Krengel, Ergodic Theorems, de Gruyter Studies in Mathematics, Vol. 6, Walter de Gruyter, Berlin, 1985.Google Scholar
  12. [12]
    J. Kurzweil, On the metric theory of inhomogeneous diophantine approximation, Studia Mathematica 15 (1955), 84–112.MathSciNetCrossRefzbMATHGoogle Scholar
  13. [13]
    L. Marchese, The Khinchin theorem for interval exchange transformations, Journal of Modern Dynamics 5 (2011), 123–183.MathSciNetCrossRefzbMATHGoogle Scholar
  14. [14]
    S. Marmi, P. Moussa and J.-C. Yoccoz, The cohomological equation for Roth-type interval exchange maps, Journal of the American Mathematical Society 18 (2005), 823–872.MathSciNetCrossRefzbMATHGoogle Scholar
  15. [15]
    H. Masur, Hausdorff dimension of the set of nonergodic foliations of a quadratic differential, Duke Mathematical Journal 66 (1992), 387–442.MathSciNetCrossRefzbMATHGoogle Scholar
  16. [16]
    W. Philipp, Some metrical theorems in number theory, Pacific Journal of Mathematics 20 (1967), 109–127.MathSciNetCrossRefzbMATHGoogle Scholar
  17. [17]
    W. Veech, Boshernitzan’s criterion for unique ergodicity of an interval exchange transformation, Ergodic Theory and Dynamical Systems 7 (1987), 149–153.MathSciNetCrossRefzbMATHGoogle Scholar
  18. [18]
    A. Zorich, Deviation for interval exchange transformations, Ergodic Theory and Dynamical Systems 17 (1997), 1477–1499.MathSciNetCrossRefzbMATHGoogle Scholar

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© Hebrew University of Jerusalem 2018

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of UtahSalt Lake CityUSA
  2. 2.Department of Mathematics and Computer ScienceWesleyan UniversityMiddletownUSA

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