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

Christopher P. Porter

TL;DR

This paper relates 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 the tree, drawing on work by Abram and Lagarias on path sets.

Abstract

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.

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

TL;DR

This paper relates 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 the tree, drawing on work by Abram and Lagarias on path sets.

Abstract

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.
Paper Structure (13 sections, 7 theorems, 44 equations, 1 figure)

This paper contains 13 sections, 7 theorems, 44 equations, 1 figure.

Table of Contents

  1. Opening
  2. Background
  3. The bigger picture
  4. Coding with unequal costs and self-similar trees
  5. Unequal costs coding
  6. Self-similar fractal trees
  7. Relating length functions and self-similarity
  8. Deriving the Lutz/Mayordomo formulas for self-similar trees
  9. A dynamical systems perspective on self-similar fractal trees
  10. Self-similar trees and path sets
  11. Spectral analysis of the pointed graph associated with a finite tree
  12. Restricting the Parry measure on a sofic shift
  13. Acknowledgements

Key Result

Theorem 4.1

Let $T\subseteq\Sigma_m^*$ be a finite tree for some $m\in\mathbb{N}$. Let $\ell_T$ be the length function induced by $T$ and $F_T$ the self-similar fractal generated by $T$. Then the channel capacity of $\ell_T$ is equal to $\log(m)\cdot\mathrm{sdim}(F)$. In particular, in the case that $m=2$, the

Figures (1)

  • Figure 1: A finite tree $T$ and its corresponding pointed directed graph $G$

Theorems & Definitions (14)

  • Theorem 4.1
  • proof
  • Theorem 4.2
  • proof
  • Theorem 5.1
  • proof
  • Theorem 5.2
  • proof
  • Theorem 5.3
  • proof
  • ...and 4 more