Dimensions of Self-affine Sets: A Survey

  • Kenneth Falconer
Part of the Trends in Mathematics book series (TM)


Self-affine sets may be expressed as unions of reduced scale affine copies of themselves. We survey general and specific constructions of self-affine sets and in particular the problem of finding or estimating their Hausdorff or box-counting dimensions. The structure and dimensional properties of self-affine sets are somewhat subtle, for example, their dimensions need not vary continuously in the defining transformations.


Hausdorff Dimension Iterate Function System Multifractal Analysis Fractal Interpolation Bernoulli Measure 
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.


  1. 1.
    Barański, K.: Hausdorff dimension of the limit sets of some planar geometric constructions. Adv. Math. 210, 215–245 (2007)MathSciNetMATHCrossRefGoogle Scholar
  2. 2.
    Barański, K.: Hausdorff dimension of self-affine limit sets with an invariant direction. Discrete Contin. Dyn. Syst. 21, 1015–1023 (2008)MathSciNetMATHCrossRefGoogle Scholar
  3. 3.
    Barnsley, M.F.: Fractal functions and interpolation. Constr. Approx. 2, 303–329 (1986)MathSciNetMATHCrossRefGoogle Scholar
  4. 4.
    Barnsley, M.F.: Fractals Everywhere, 2nd edn. Academic Press Professional, Boston (1993)MATHGoogle Scholar
  5. 5.
    Barnsley, M.F., Harrington, A.N.: The calculus of fractal interpolation functions. J. Approx. Theory 57, 14–34 (1989)MathSciNetMATHCrossRefGoogle Scholar
  6. 6.
    Barral, J., Feng, D.-J.: Multifractal formalism for almost all self-affine measures, to appear, Comm. Math. Phys.Google Scholar
  7. 7.
    Barral, J., Mensi, M.: Gibbs measures on self-affine Sierpiński carpets and their singularity spectrum. Ergodic Theory Dynam. Syst. 27, 1419–1443 (2007)MathSciNetMATHCrossRefGoogle Scholar
  8. 8.
    Barreira, L: A non-additive thermodynamic formalism and applications to dimension theory of hyperbolic dynamical systems. Ergodic Theory Dynam. Syst. 16, 871–927 (1996)MathSciNetMATHCrossRefGoogle Scholar
  9. 9.
    Barreira, L: Dimension estimates in nonconformal hyperbolic dynamics. Nonlinearity 16, 1657–1672 (2003)MathSciNetMATHCrossRefGoogle Scholar
  10. 10.
    Barreira, L.: Thermodynamic Formalism and Applications to Dimension Theory. Birkhäuser, Basel (2011)MATHCrossRefGoogle Scholar
  11. 11.
    Barreira, L., Gelfert, K.: Dimension estimates in smooth dynamics: a survey of recent results. Ergodic Theory Dynam. Syst. 31, 641–671 (2011)MathSciNetMATHCrossRefGoogle Scholar
  12. 12.
    Bedford, T.: Crinkly curves, Markov partitions and box dimensions in self-similar sets. PhD thesis, University of Warwick (1984)Google Scholar
  13. 13.
    Bedford, T.: The box dimension of self-affine graphs and repellers. Nonlinearity 2, 53–71 (1989)MathSciNetMATHCrossRefGoogle Scholar
  14. 14.
    Bedford, T., Urbański, M.: The box and Hausdorff dimension of self-affine sets. Ergodic Theory Dynam. Syst. 10, 627–644 (1990)MATHCrossRefGoogle Scholar
  15. 15.
    Bowen, R.: Hausdorff dimension of quasi-circles. Publ. Math. IHES 50, 11–26 (1979)MathSciNetMATHGoogle Scholar
  16. 16.
    Chen, J., Pesin, Y.: Dimension of non-conformal repellers: a survey. Nonlinearity 23, R93–R114 (2010)MathSciNetMATHCrossRefGoogle Scholar
  17. 17.
    Douady, A., Oesterlé, J.: Dimension de Hausdorff des attracteurs. C. R. Acad. Sci. Paris Sr. A 290, 1135–1138 (1980)Google Scholar
  18. 18.
    Edgar, G.A.: Fractal dimension of self-affine sets: some examples. Rend. Circ. Mat. Palermo (2) Suppl. 28, 341–358 (1988)Google Scholar
  19. 19.
    Falconer, K.J.: The Hausdorff dimension of self-affine fractals. Math. Proc. Cambridge Philos. Soc. 103, 339–350 (1988)MathSciNetMATHCrossRefGoogle Scholar
  20. 20.
    Falconer, K.J.: The dimension of self-affine fractals II. Math. Proc. Cambridge Philos. Soc. 111, 169–179 (1992)MathSciNetMATHCrossRefGoogle Scholar
  21. 21.
    Falconer, K.J.: Bounded distortion and dimension for non-conformal repellers. Math. Proc. Cambridge Philos. Soc. 115, 315–334 (1994)MathSciNetMATHCrossRefGoogle Scholar
  22. 22.
    Falconer, K.J.: Techniques in Fractal Geometry. Wiley, Chichester (1997)MATHGoogle Scholar
  23. 23.
    Falconer, K.J.: Generalized dimensions of measures on self-affine sets. Nonlinearity 12, 877–891 (1999)MathSciNetMATHCrossRefGoogle Scholar
  24. 24.
    Falconer, K.J.: Fractal Geometry—Mathematical Foundations and Applications, 2nd edn. Wiley, Chichester (2003)MATHCrossRefGoogle Scholar
  25. 25.
    Falconer, K.J.: Generalised dimensions of measures on almost self-affine sets. Nonlinearity 23, 1047–1069 (2010)MathSciNetMATHCrossRefGoogle Scholar
  26. 26.
    Falconer, K.J., Lammering, B.: Fractal properties of general Sirepiński triangles. Fractals 6, 31–41 (1998)MathSciNetMATHCrossRefGoogle Scholar
  27. 27.
    Falconer, K.J., Miao, J.: Dimensions of self-affine fractals and multifractals generated by upper-triangular matrices. Fractals 15, 289–299 (2007)MathSciNetMATHCrossRefGoogle Scholar
  28. 28.
    Falconer, K.J., Miao, J.: Exceptional sets for self-affine fractals. Math. Proc. Cambridge Philos. Soc. 145, 669–684 (2008)MathSciNetMATHCrossRefGoogle Scholar
  29. 29.
    Falconer, K.J., Sloan, A.: Continuity of subadditive pressure for self-affine sets. R. Anal. Exchange 34, 413–428 (2009)MathSciNetMATHGoogle Scholar
  30. 30.
    Feng, D.-J., Wang, Y.: A class of self-affine sets and self-affine measures. J. Fourier Anal. Appl. 11, 107–124 (2005)MathSciNetMATHCrossRefGoogle Scholar
  31. 31.
    Fraser, J.M.: On the packing dimension of box-like self-affine sets in the plane, Nonlinearity 25, 2075–2092 (2012)MathSciNetMATHCrossRefGoogle Scholar
  32. 32.
    Gatzouras, D., Lalley, S.P.: Hausdorff and box dimensions of certain self-affine fractals. Indiana Univ. Math. J. 41, 533–568 (1992)MathSciNetMATHCrossRefGoogle Scholar
  33. 33.
    Gatzouras, D., Peres, Y.: Invariant measures of full dimension for some expanding maps. Ergodic Theory Dynam. Syst. 17, 147–67 (1997)MathSciNetMATHCrossRefGoogle Scholar
  34. 34.
    Hueter, I., Lalley, S.P.: Falconer’s formula for the Hausdorff dimension of a self-affine set in ℝ 2. Ergodic Theory Dynam. Syst. 15, 77–97 (1995)MathSciNetMATHCrossRefGoogle Scholar
  35. 35.
    Hutchinson, J.E.: Fractals and self-similarity. Indiana Univ. Math. J. 30, 713–747 (1981)MathSciNetMATHCrossRefGoogle Scholar
  36. 36.
    Jordan, T., Pollicott, M., Simon, K.: Hausdorff dimension for randomly perturbed self affine attractors. Commun. Math. Phys. 270, 519–544 (2007)MathSciNetMATHCrossRefGoogle Scholar
  37. 37.
    Jordan, T., Rams, M.: Multifractal analysis for Bedford-McMullen carpets. Math. Proc. Cambridge Philos. Soc. 150, 147–156 (2011)MathSciNetMATHCrossRefGoogle Scholar
  38. 38.
    Kenyon, R., Peres, Y.: Hausdorff dimensions of sofic affine-invariant sets. Israel J. Math. 94, 157–178 (1996)MathSciNetCrossRefGoogle Scholar
  39. 39.
    King, J.F.: The singularity spectrum for general Sierpiński carpets. Adv. Math. 116, 1–11 (1995)MathSciNetMATHCrossRefGoogle Scholar
  40. 40.
    Käenmäki, A., Shmerkin, P.: Overlapping self-affine sets of Kakeya type. Ergodic Theory Dynam. Syst. 29, 941–965 (2009)MATHCrossRefGoogle Scholar
  41. 41.
    Käenmäki, A., Vilppolainen, M.: Dimension and measures on sub-self-affine sets. Monatsh. Math. 161, 271–293 (2010)MathSciNetMATHCrossRefGoogle Scholar
  42. 42.
    Kono, N.: On self-affine functions. Japan J. Appl. Math. 3, 259–269 (1986)MathSciNetMATHCrossRefGoogle Scholar
  43. 43.
    Ledrappier, F.: On the dimension of some graphs. In: Symbolic Dynamics and Its Applications. Contemp. Math. vol. 135, pp. 285–293. Amer. Math. Soc., Providence (1992)Google Scholar
  44. 44.
    Luzia, N.: Hausdorff dimension for an open class of repellers in ℝ 2. Nonlinearity 19, 2895–2908 (2006)MathSciNetMATHCrossRefGoogle Scholar
  45. 45.
    Manning, A., Simon, K.: Subadditive pressure for triangular maps. Nonlinearity 20, 133–149 (2007)MathSciNetMATHCrossRefGoogle Scholar
  46. 46.
    McMullen, C.: The Hausdorff dimension of general Sierpiński carpets. Nagoya Math. J. 96, 1–9 (1984)MathSciNetMATHGoogle Scholar
  47. 47.
    Olsen, L.: Self-affine multifractal Sierpiński sponges in ℝ d. Pacific J. Math. 183, 143–199 (1998)MathSciNetMATHCrossRefGoogle Scholar
  48. 48.
    Olsen, L.: Symbolic and geometric local dimensions of self-affine multifractal Sierpiński sponges in ℝ d. Stoch. Dyn. 7, 37–51 (2007)MathSciNetMATHCrossRefGoogle Scholar
  49. 49.
    Peres, Y., Solomyak, B.: Problems on self-similar sets and self-affine sets: an update. In: Bandt, C., Graf, S., Zähle, M. (eds.) Fractal Geometry and Stochastics II. Progress in Probability, vol. 46, pp. 95–106. Birkhäuser, Basel (2000)CrossRefGoogle Scholar
  50. 50.
    Pesin, Y.B.: Dimension Theory in Dynamical Systems. University of Chicago Press, Chicago (1997)Google Scholar
  51. 51.
    Pollicott, M., Weiss, H.: The dimensions of some self-affine limit sets in the plane. J. Stat. Phys. 77 (1994), 841–866.MathSciNetMATHCrossRefGoogle Scholar
  52. 52.
    Robinson, J.C.: Dimensions, Embeddings, and Attractors. Cambridge Tracts in Mathematics, vol. 186. Cambridge University Press, Cambridge (2011)Google Scholar
  53. 53.
    Solomyak, B.: Measure and dimensions for some fractal families. Math. Proc. Cambridge Philos. Soc. 124, 531–546 (1998)MathSciNetMATHCrossRefGoogle Scholar
  54. 54.
    Shmerkin, P.: Overlapping self-affine sets. Indiana Univ. Math. J. 55, 1291–1331 (2006)MathSciNetMATHCrossRefGoogle Scholar
  55. 55.
    Temam, R.: Infinite-dimensional dynamical systems in mechanics and physics. Applied Mathematical Sciences, vol. 68, 2nd edn. Springer, New York (1997)Google Scholar
  56. 56.
    Urbański, M.: The probability distribution and Hausdorff dimension of self-affine functions. Probab. Theory Related Fields 84, 377–391 (1990)MathSciNetMATHCrossRefGoogle Scholar
  57. 57.
    Urbański, M.: The Hausdorff dimension of the graphs of continuous self-affine functions. Proc. Amer. Math. Soc. 108, 921–930 (1990)MathSciNetMATHGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2013

Authors and Affiliations

  1. 1.Mathematical InstituteUniversity of St AndrewsSt AndrewsScotland

Personalised recommendations