Numeration Systems and Markov Partitions from Self Similar Tilings

Using self similar tilings we represent the elements of Rn as digit expansions with digits in Rn being operated on by powers of an expansive linear map. We construct Markov partitions for hyperbolic toral automorphisms by considering a special class of self similar tilings modulo the integer lattice. We use the digit expansions inherited from these tilings to give a symbolic representation for the toral automorphisms. Fractals and fractal tilings have captured the imaginations of a wide spectrum of disciplines. Computer generated images of fractal sets are displayed in public science centers, museums, and on the covers of scientific journals. Fractal tilings which have interesting properties are finding applications in many areas of mathematics. For example, number theorists have linked fractal tilings of R with numeration systems for R in complex bases [16], [8]. We will see that fractal self similar tilings of R provide natural building blocks for numeration systems of R. These numeration systems generalize the 1-dimensional cases in [14],[10],[11] as well as the 2-dimensional cases mentioned above. Our motivation for studying fractal tilings comes from ergodic theory. In [2] Adler and Weiss show that topological entropy is a complete invariant for metric equivalence of continuous ergodic automorphisms of the two-dimensional torus. Their method of proof is to construct a partition of the 2-torus which satisfies certain properties. The partition is called a Markov partition. By assigning each element of the partition a symbol, it is possible to assign each point in the 2-torus a bi-infinite sequence of symbols which corresponds to the orbit of the point. The objective is to represent the continuous dynamical system as a symbolic one in such a way that periodicity and transitivity is preserved in the representation. In [4] Bowen shows that every Anosov diffeomorphism has a Markov partition. In particular every hyperbolic toral automorphism has a Markov partition. His construction uses a recursive definition which deforms rectangles in the stable and unstable directions. While existence is shown, the proof does not indicate an efficient way to actually construct the partitions. In [5] Bowen shows that in the case of the 3-torus the boundary sets of the Markov partition have fractional Hausdorff dimension. In [3] Bedford constructs examples of Markov partitions for the 3-torus and describes the sets as crinkly tin cans. In [6] Cawley generalizes Bowen’s results to higher dimensional tori. Since Markov partitions have properties which resemble those of self similar tilings and the boundary sets of these partitions have fractional dimension, it is Received by the editors October 2, 1996. 1991 Mathematics Subject Classification. Primary 58F03, 34C35, 54H20. c ©1999 American Mathematical Society