Hausdorff dimensions of sofic affine-invariant sets

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.

[1]  T. Kamae.  A characterization of self-affine functions , 1986 .

[2]  The Hausdorff dimension of the graphs of continuous self-affine functions , 1990 .

[3]  H. Kesten.  Random difference equations and Renewal theory for products of random matrices , 1973 .

[4]  N. Kôno.  On self-affine functions , 1986 .

[5]  Curtis T. McMullen,et al.  The Hausdorff dimension of general Sierpiński carpets , 1984, Nagoya Mathematical Journal.

[6]  Kenneth Falconer,et al.  The Hausdorff dimension of self-affine fractals , 1988, Mathematical Proceedings of the Cambridge Philosophical Society.

[7]  Tim Bedford,et al.  The box and Hausdorff dimension of self-affine sets , 1990, Ergodic Theory and Dynamical Systems.

[8]  E. L. Page.  Théorèmes de renouvellement pour les produits de matrices aléatoires Équations aux différences aléatoires , 1983 .

[9]  Mike Boyle,et al.  A note on minimal covers for sofic systems , 1985 .

[10]  D. Hardin,et al.  Dimensions associated with recurrent self-similar sets , 1991, Mathematical Proceedings of the Cambridge Philosophical Society.

[11]  L. Young.  Dimension, entropy and Lyapunov exponents , 1982, Ergodic Theory and Dynamical Systems.

[12]  Kenneth Falconer,et al.  Fractal Geometry: Mathematical Foundations and Applications , 1990 .

[13]  T. Bedford.  On Weierstrass-like functions and random recurrent sets , 1989, Mathematical Proceedings of the Cambridge Philosophical Society.

[14]  Yuval Peres,et al.  Intersecting random translates of invariant Cantor sets , 1991 .

[15]  Kenneth Falconer,et al.  The dimension of self-affine fractals II , 1992, Mathematical Proceedings of the Cambridge Philosophical Society.

[16]  B. Kitchens,et al.  Boundaries of Markov partitions , 1992 .

[17]  R. Kaufman.  On Hausdorff dimension of projections , 1968 .

[18]  W. Parry.  Intrinsic Markov chains , 1964 .

[19]  The probability distribution and Hausdorff dimension of self-affine functions , 1990 .

[20]  P. Walters.  Introduction to Ergodic Theory , 1977 .

[21]  T. Bedford.  The box dimension of self-affine graphs and repellers , 1989 .

[22]  Tim Bedford,et al.  Generating special Markov partitions for hyperbolic toral automorphisms using fractals , 1986, Ergodic Theory and Dynamical Systems.

[23]  Steven P. Lalley,et al.  Hausdorff and box dimensions of certain self-affine fractals , 1992 .

[24]  P. Billingsley,et al.  Ergodic theory and information , 1966 .

[25]  B. Weiss.  Subshifts of finite type and sofic systems , 1973 .

[26]  R. Bowen.  Equilibrium States and the Ergodic Theory of Anosov Diffeomorphisms , 1975 .

[27]  R. Daniel Mauldin,et al.  Hausdorff dimension in graph directed constructions , 1988 .