Symbolic dynamics: entropy = dimension = complexity

TL;DR

This paper establishes a fundamental equivalence between entropy, Hausdorff dimension, and Kolmogorov complexity for subshifts over multi-dimensional groups, revealing deep connections between dynamical, geometric, and computational complexity.

Contribution

It proves that for subshifts over  groups, topological entropy equals Hausdorff dimension and characterizes this equivalence via Kolmogorov complexity, providing a unified view of complexity measures.

Findings

Entropy equals Hausdorff dimension for subshifts over  groups.

Characterization of entropy and dimension through Kolmogorov complexity.

Clarification and correction of a typographical error in the published proof.

Abstract

Let $G$ be the group $\mathbb{Z}^d$ or the monoid $\mathbb{N}^d$ where $d$ is a positive integer. Let $X$ be a subshift over $G$, i.e., a closed and shift-invariant subset of $A^G$ where $A$ is a finite alphabet. We prove that the topological entropy of $X$ is equal to the Hausdorff dimension of $X$ and has a sharp characterization in terms of the Kolmogorov complexity of finite pieces of the orbits of $X$. In the version of this paper that has been published in Theory of Computing Systems, the proof of Lemma 4.3 contains a confusing typographical error. This version of the paper corrects that error.