Information vs Dimension -- an Algorithmic Perspective

TL;DR

This paper reviews the relationship between fractal dimensions and algorithmic information theory, exploring the development of Kolmogorov complexity, Hausdorff measures, and the informal identity linking entropy, complexity, and dimension, with new insights on multifractal measures.

Contribution

It develops the informal identity 'entropy = complexity = dimension' from first principles and offers new observations on multifractal measures from an algorithmic perspective.

Findings

Clarifies the connection between fractal dimensions and Kolmogorov complexity.

Provides new insights into multifractal measures using algorithmic information theory.

Synthesizes 30 years of research on the topic.

Abstract

This paper surveys work on the relation between fractal dimensions and algorithmic information theory over the past thirty years. It covers the basic development of prefix-free Kolmogorov complexity from an information theoretic point of view, before introducing Hausdorff measures and dimension along with some important examples. The main goal of the paper is to motivate and develop the informal identity "entropy = complexity = dimension" from first principles. The last section of the paper presents some new observations on multifractal measures from an algorithmic viewpoint.