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Israel Journal of Mathematics

, Volume 94, Issue 1, pp 157–178 | Cite as

Hausdorff dimensions of sofic affine-invariant sets

  • R. Kenyon
  • Y. Peres
Article

Abstract

We determine the Hausdorff and Minkowski dimensions of compact subsets of the 2-torus which are invariant under a linear endomorphism with integer eigenvalues and correspond to shifts of finite type or sofic shifts via some Markov partition. This extends a result of McMullen (1984) and Bedford (1984), who considered full-shifts.

Keywords

Spectral Radius Hausdorff Dimension Finite Type Topological Entropy Sierpinski Gasket 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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References

  1. [1]
    J. Ashley, B. Kitchens and M. Stafford,Boundaries of Markov partitions, Transactions of the American Mathematical Society333 (1992), 177–202.MATHCrossRefMathSciNetGoogle Scholar
  2. [2]
    T. Bedford,Crinkly curves, Markov partitions and box dimension in self-similar sets, Ph.D. Thesis, University of Warwick, 1984.Google Scholar
  3. [3]
    T. Bedford,Generating special Markov partitions for hyperbolic toral automorphisms using fractals, Ergodic Theory and Dynamical Systems6 (1986), 325–333.MATHMathSciNetGoogle Scholar
  4. [4]
    T. Bedford,The box dimension of self-affine graphs and repellers, Nonlinearity2 (1989), 53–71.MATHCrossRefMathSciNetGoogle Scholar
  5. [5]
    T. Bedford,On Weierstrass-like functions and random recurrent sets, Mathematical Proceedings of the Cambridge Philosophical Society106 (1989), 325–342.MATHMathSciNetGoogle Scholar
  6. [6]
    T. Bedford and M. Urbanski,The box and Hausdorff dimension of self-affine sets, Ergodic Theory and Dynamical Systems10 (1990), 627–644.MATHMathSciNetGoogle Scholar
  7. [7]
    P. Billingsley,Ergodic Theory and Information, Wiley, New York, 1965.MATHGoogle Scholar
  8. [8]
    P. Bougerol and J. Lacroix,Products of Random Matrices with Applications to Schroedinger Operators, Birkhauser, Boston, 1985.MATHGoogle Scholar
  9. [9]
    R. Bowen,Equilibrium states and the ergodic theory of Anosov diffeomorphisms, Lecture Notes in Mathematics, Vol. 470, Springer-Verlag, Berlin, 1974.Google Scholar
  10. [10]
    M. Boyle, B. Kitchens and B. Marcus,A note on minimal covers for sofic systems, Proceedings of the American Mathematical Society95 (1985), 403–411.MATHCrossRefMathSciNetGoogle Scholar
  11. [11]
    F. M. Dekking,Recurrent sets, Advances in Mathematics44 (1982), 78–104.MATHCrossRefMathSciNetGoogle Scholar
  12. [12]
    A. Deliu, J. S. Geronimo, R. Shonkwiler and D. Hardin,Dimensions associated with recurrent self-similar sets, Mathematical Proceedings of the Cambridge Philosophical Society110 (1991), 327–336.MATHMathSciNetGoogle Scholar
  13. [13]
    K. J. Falconer,The Hausdorff dimension of self-affine fractals, Mathematical Proceedings of the Cambridge Philosophical Society103 (1988), 339–350.MATHMathSciNetGoogle Scholar
  14. [14]
    K. J. Falconer,Fractal Geometry—Mathematical Foundations and Applications, Wiley, New York, 1990.MATHGoogle Scholar
  15. [15]
    K. J. Falconer,The dimension of self-affine fractals II, Mathematical Proceedings of the Cambridge Philosophical Society111 (1992), 169–179.MATHMathSciNetCrossRefGoogle Scholar
  16. [16]
    H. Furstenberg,Disjointness in ergodic theory, minimal sets and a problem in Diophantine approximation, Mathematical Systems Theory1 (1967), 1–49.MATHCrossRefMathSciNetGoogle Scholar
  17. [17]
    D. Gatzouras and S. Lalley,Hausdorff and box dimensions of certain self-affine fractals, Indiana University Mathematics Journal41 (1992) 533–568.MATHCrossRefMathSciNetGoogle Scholar
  18. [18]
    T. Kamae,A characterization of self-affine functions, Japan Journal of Applied Mathematics3 (1986), 271–280.MATHMathSciNetGoogle Scholar
  19. [19]
    R. Kaufman,On Hausdorff dimension of projections, Mathematika15 (1968), 153–155.MathSciNetMATHCrossRefGoogle Scholar
  20. [20]
    R. Kenyon and Y. Peres,Intersecting random translates of invariant Cantor sets, Inventiones mathematicae104 (1991), 601–629.MATHCrossRefMathSciNetGoogle Scholar
  21. [21]
    H. Kesten,Random difference equations and renewal theory for products of random matrices, Acta Mathematica131 (1973), 207–248.MATHCrossRefMathSciNetGoogle Scholar
  22. [22]
    N. Kono,On self-affine functions, Japan Journal of Applied Mathematics3 (1986), 259–269.MATHMathSciNetCrossRefGoogle Scholar
  23. [23]
    E. Lepage,Théorèmes de renouvellement pour les produits de matrices aléatoires; Equations aux differences aléatoires, Séminaire de Probabilités, Rennes, IRMAR, 1983.Google Scholar
  24. [24]
    R. D. Mauldin and S. C. Williams,Hausdorff dimension in graph directed constructions, Transactions of the American Mathematical Society309 (1988), 811–829.MATHCrossRefMathSciNetGoogle Scholar
  25. [25]
    C. McMullen,The Hausdorff dimension of general Sierpinski carpets, Nagoya Mathematical Journal96 (1984), 1–9.MATHMathSciNetGoogle Scholar
  26. [26]
    W. Parry,Intrinsic Markov chains, Transactions of the American Mathematical Society,112 (1964), 55–65.MATHCrossRefMathSciNetGoogle Scholar
  27. [27]
    M. Urbanski,The Hausdorff dimension of the graphs of continuous self-affine functions, Proceedings of the American Mathematical Society108 (1990), 921–930.MATHCrossRefMathSciNetGoogle Scholar
  28. [28]
    M. Urbanski,The probability distribution and Hausdorff dimension of self-affine functions, Probability Theory and Related Fields84 (1990), 377–391.MATHCrossRefMathSciNetGoogle Scholar
  29. [29]
    P. Walters,An Introduction to Ergodic Theory, Springer-Verlag, Berlin, 1982.MATHGoogle Scholar
  30. [30]
    B. Weiss,Subshifts of finite type and sofic systems, Monatshefte für Mathematik77 (1973), 462–474.MATHCrossRefGoogle Scholar
  31. [31]
    L. S. Young,Dimension, entropy and Lyapunov exponents, Ergodic Theory and Dynamical Systems2 (1982), 109–124.MATHMathSciNetCrossRefGoogle Scholar

Copyright information

© The Magnes Press 1996

Authors and Affiliations

  • R. Kenyon
    • 1
  • Y. Peres
    • 2
  1. 1.CNRS UMR 128Ecole Normale Supérieure de LyonLyonFrance
  2. 2.Department of StatisticsUniversity of California at BerkeleyBerkeleyUSA

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