Abstract
In this article we compute the Hausdorff dimension and box dimension (or capacity) of a dynamically constructed model similarity process in the plane with two distinct contraction coefficients. These examples are natural generalizations to the plane of the simple Markov map constructions for Cantor sets on the line. Some related problems have been studied by different authors; however, those results are directed toward generic results in quite general situations. This paper concentrates on computing explicit formulas in as many specific cases as possible. The techniques of previous authors and ours are correspondingly very different. In our calculations, delicate number-theoretic properties of the contraction coefficients arise. Finally, we utilize the results for the model problem to compute the dimensions of some affine horseshoes in ℝn, and we observe that the dimensions do not always coincide and their coincidence depends on delicate number-theoretic properties of the Lyapunov exponents.
Similar content being viewed by others
References
J. Alexander and J. Yorke, Fat baker's transformations,Ergod. Theory Dynam. Syst. 4:1–23 (1984).
T. Bedford, Crinkly curves, Markov partitions and box dimension in self-similar sets, Ph.D. thesis, Warwick (1984).
T. Bedford and M. Urbanski, The box and Hausdorff dimension of self-affine sets,Ergod. Theory Dynam. Syst. 10:627–644 (1990).
P. Erdos, On the smoothness properties of a family of Bernoulli convolutions,Am. J. Math. 1962:180–186.
K. Falconer,The Geometry of Fractal Sets (Cambridge University Press, Cambridge, 1985).
K. Falconer,Fractal Geometry, Mathematical Foundations and Applications (Cambridge University Press, Cambridge, 1990).
K. Falconer, The Hausdorff dimension of self-affine fractals I,Math. Proc. Camb. Phil. Soc. 103:339–350 (1988).
K. Falconer, The Hausdorff dimension of self-affine fractals II.Math. Proc. Camb. Phil. Soc. 111:169–179 (1992).
A. Garsia, Arithmetic properties of Bernoulli convolutions,Trans. Am. Math. Soc. 102:409–432 (1962).
R. Kenyon and Y. Peres, Hausdorff dimensions of affine-invariant sets and Sierpinski, preprint (1992).
J. Kahane and R. Salem, Sur la convolution d'une infinite de distributions de Bernoulli,Colloq. Math. VI:193–202 (1958).
F. Ledrappier, On the dimension of some graphs, preprint (1992).
S. Lalley and D. Gatzouras, Hausdorff and box dimensions of certain self-affine fractals,Indiana J. Math. 41:533–568 (1992).
C. McMullen, The Hausdorff dimension of general Sierpiński carpets,Nagoya Math. J. 96:1–9 (1984).
W. Parry and M. Pollicott, Zeta functions and the periodic orbit structure of hyperbolic dynamics,Asterisque 187–188:1–268 (1990).
F. Przytycki and M. Urbański, On Hausdorff dimension of some fractal sets,Studia Math. 93:155–186 (1989).
D. Ruelle, Spectral properties of a class of operators associated with maps in one dimension.Ergod. Theory Dynam. Syst. 11:757–767 (1991).
M. Reed and B. Simon,Functional Analysis, Vol. I (Academic Press, 1980).
R. Salem,Algebraic Numbers and Fourier Analysis (Health, 1963).
P. Walters,Introduction to Ergodic Theory (Springer-Verlag, 1982).
L. S. Young, Dimension, entropy, and Lyapunov exponents,Ergod. Theory Dynam. Syst. 2:109–124 (1982).
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Pollicott, M., Weiss, H. The dimensions of some self-affine limit sets in the plane and hyperbolic sets. J Stat Phys 77, 841–866 (1994). https://doi.org/10.1007/BF02179463
Received:
Accepted:
Issue Date:
DOI: https://doi.org/10.1007/BF02179463