Subfractals Induced by Subshifts

Abstract

In this thesis, a subfractal is the subset of points in the attractor of an iterated function system in which every point in the subfractal is associated with an allowable word from a subshift on the underlying symbolic space. In the case in which (1) the subshift is a subshift of nite type with an irreducible adjacency matrix, (2) the iterated function system satis es the open set condition, and (3) contractive bounds exist for each map in the iterated function system, we nd bounds for both the Hausdor and box dimensions of the subfractal, where the bounds depend both on the adjacency matrix and the contractive bounds on the maps. We extend this result to so c subshifts, a more general subshift than a subshift of nite type, and to allow the adjacency matrix to be reducible. The structure of a subfractal naturally de nes a measure on Rn. For an iterated function system which satis es the open set condition and in which the maps are similitudes, we construct an invariant measure supported on a subfractal induced by a subshift of nite type. For this speci c measure, we calculate the local dimension for almost every point, and hence calculate the Hausdor dimension for the measure.