The exact Hausdorff dimension for a class of fractal functions

Abstract

In recent years a number of authors (an incomplete list is [ 1,2, 3,6, 7, 9, 11, 12, 131) have considered a class of real functions whose graphs are, in general, fractal sets in R2. In this paper we give sufficient conditions for the fractal and Hausdorff dimensions to be equal for a certain subclass of fractal functions. The sets we consider are examples of self-affine fractals generated using iterated function systems (i.f.s.). Falconer [S] has shown that for almost all such sets the fractal and Hausdorff dimensions are equal and he gives a formula for the common dimension, due originally to Moran [S]. These results, however, give no information about individual fractal functions, In this paper we extend Moran’s original method and show that if certain conditions on the i.f.s. are satisfied, then the two dimensions are equal. Kono [ 111 and Bedford [ 121 considered special cases of the subclass of fractal functions that we will introduce. Bedford and Urbanski [13] use a nonlinear setting to present conditions for the equality of Hausdorff and fractal dimension. However, their criteria are based on measure-theoretic characterizations and the use of the concept of generalized pressure. Our criterion on the other hand is based on the underlying geometry of the attractor and is easier to verify. We will show this on two specific examples which are more general than the self-ahine functions presented in [13].