# Constant length substitutions, iterated function systems and amorphic complexity

## TL;DR

This paper uses fractal geometry and iterated function systems to analyze amorphic complexity in symbolic dynamics, providing bounds and formulas especially for constant length substitutions with binary alphabets.

## Contribution

It introduces a geometric, dimensional approach to characterize amorphic complexity for subshifts with discrete spectrum, extending the understanding of zero entropy systems.

## Key findings

- Established a dimensional characterization of amorphic complexity.
- Derived bounds for amorphic complexity using iterated function systems.
- Provided a closed formula for binary alphabet cases.

## Abstract

We show how geometric methods from the general theory of fractal dimensions and iterated function systems can be deployed to study symbolic dynamics in the zero entropy regime. More precisely, we establish a dimensional characterization of the topological notion of amorphic complexity. For subshifts with discrete spectrum associated to constant length substitutions, this characterization allows us to derive bounds for the amorphic complexity by interpreting the subshift as the attractor of an iterated function system in a suitable quotient space. As a result, we obtain the general finiteness and positivity of amorphic complexity in this setting and provide a closed formula in case of a binary alphabet.