Length Functions and the Dimension of Points in Self-Similar Fractal Trees

In this paper, we study the effective dimension of points in infinite fractal trees generated recursively by a finite tree over some alphabet. Using unequal costs coding, we associate a length function with each such fractal tree and show that the channel capacity of the length function is equal to the similarity dimension of the fractal tree (up to a multiplicative constant determined by the size of the alphabet over which our tree is defined). Using this result, we derive formulas for calculating the effective dimension and strong effective dimension of points in fractal trees, establishing analogues of several results due to Lutz and Mayordomo, who studied the effective dimension of points in self-similar fractals in Euclidean space. Lastly, we explore the connections between the channel capacity of a length function derived from a finite tree and the measure of maximum entropy on a related directed multigraph that encodes the structure of our tree, drawing on work by Abram and Lagarias on path sets, where a path set is a generalization of the notion of a sofic shift.