# Hausdorff dimensions for graph-directed measures driven by infinite rooted trees

## TL;DR

This paper establishes bounds for the Hausdorff dimensions of graph-directed measures on infinite N-ary trees, encompassing measures from fractal geometry and functional equations, extending beyond finite graph cases.

## Contribution

It provides new bounds for Hausdorff dimensions of measures on infinite trees, including non-Gibbs and non-self-similar measures relevant to fractal geometry.

## Key findings

- Derived upper and lower bounds for Hausdorff dimensions.
- Included measures from harmonic functions and de Rham's equations.
- Extended analysis to infinite N-ary tree structures.

## 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 Sierp\'inski gasket, the Kusuoka energy measures on it, and, measures defined by solutions of de Rham's functional equations driven by linear fractional transformations.