Hausdorff Dimensions for Graph-directed Measures Driven by Infinite Rooted Trees

Abstract

We give upper and lower bounds for the Hausdorff dimensions for a class of graph-directed measures when its underlying directed graph is the infinite \(N\)-ary tree. These measures are different from graph-directed self-similar measures driven by finite directed graphs and are not necessarily Gibbs measures. However our class contains several measures appearing in fractal geometry and functional equations, specifically, measures defined by restrictions of non-constant harmonic functions on the two-dimensional Sierpínski gasket, the Kusuoka energy measures on it, and, measures defined by solutions of de Rham’s functional equations driven by linear fractional transformations.