Dimension and Dynamics for Fractal Recurrent Sets

Abstract

The fractal 'recurrent sets' defined by F. M. Dekking are analysed using subshifts of finite type. We show how Dekking's method is related to a construction due to J. Hutchinson, and prove a conjecture of Dekking concerning conditions under which the best general upper bound for the Hausdorff dimension for recurrent sets is actually equal to the Hausdorff dimension. Introduction Since the publication of Mandelbrot's book [9] there has been wide interest in fractals. The appearence of [9] stimulated several mathematical papers, including one by Hutchinson [8] which analysed strictly self-similar sets using constructions closely related to the full shift spaces and Markov partitions used in the study of dynamical systems. A different construction of fractals was given by Dekking [5]. Here we show how Dekking's method is related to that of Hutchinson. There is a best general upper estimate of Hausdorff dimension for recurrent sets, and we are interested in finding conditions under which this general estimate is actually equal to the Hausdorff dimension. When the scaling map is a similitude, this depends on how the different pieces of the fine structure of a recurrent set intersect. Dekking [7] gives a condition called 'resolvability' which ensures that such intersection is only slight. We prove his conjecture, that resolvability holds if and only if the general estimate is equal to the Hausdorff dimension, by using techniques from dynamical systems. Dekking's formalism is useful for constructing mathematical objects because it gives a large degree of control over the geometric properties of the fractal to be generated. In another paper [2] we use recurrent sets to construct invariant sets and 'canonical' Markov partitions for hyperbolic automorphisms of the 3-torus. I should like to thank F. M. Dekking whose work stimulated my own and C. Series who supervised part of this work. I should also like to thank Noel Lloyd and the referee for their comments. This work was supported by the SERC and the King's College Research Centre. Background and definitions A quantity which gives a notion of the 'rarity' or 'thickness' of a set is Hausdorff dimension. Given Y c U the a-dimensional Hausdorff measure of Y is defined as HMa(Y) = lim infix |diamQ|°: {JC^Y £-»o u i Y Received 1 March 1985. 1980 Mathematics Subject Classification 28A75. J. London Math. Soc. (2) 33 (1986) 89-100