93-240 Mark Pollicott, Howard Weiss

THE DIMENSIONS OF SOME SELF AFFINE LIMIT SETS IN THE PLANE AND HYPERBOLIC SETS (61K, TeX) Oct 1, 93

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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 towards generic results in quite general situations. This paper concentrates on computing explicit formulae 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. {\bf Finally, we utilize the results for the model problem to compute the dimensions of some linear horseshoes in $ \ Bbb R^n $, and we observe that the dimensions do not always coincide and their coincidence depends on delicate number theoretic properties of the Lyapunov exponents.

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