# Scaling functions for graph directed Markov systems

## TL;DR

This paper introduces a new scaling function for graph directed Markov systems, proving it is a complete invariant for certain conjugacies and exploring its implications for the dimension theory of limit sets.

## Contribution

It defines the scaling function for these systems, proves its invariance under conjugacy, and links it to pressure and dimension theory, advancing understanding of their geometric properties.

## Key findings

- Scaling function is Hölder continuous on the dual symbolic Cantor set.
- Unique Cantor limit set exists under natural conditions.
- Scaling function fully characterizes $C^{1+eta}$ conjugacy between limit sets.

## Abstract

We introduce the scaling function associated to a graph directed Markov system, and show that it is a H\"{o}lder continuous function of the dual symbolic Cantor set. With some natural separation and regularity conditions, each such system has a unique Cantor limit set in Euclidean space. We prove that the scaling function is a complete invariant of $ C^{1+\alpha} $ conjugacy between limit sets. We conclude by relating the scaling function to the pressure, and discussing several applications to the dimension theory of limit sets.