# Complexity and invariant measure of the period-doubling subshift

## TL;DR

This paper provides direct combinatorial proofs for the complexity and invariant measure of the period-doubling subshift, deriving explicit formulas for correlation and recurrence, and analyzing its determinism as the threshold approaches zero.

## Contribution

It offers new direct proofs and explicit formulas for key properties of the period-doubling subshift, expanding understanding beyond prior indirect methods.

## Key findings

- Explicit formulas for complexity and invariant measure derived from combinatorial properties.
- Correlation integral and recurrence characteristics explicitly calculated.
- Determinism converges to 1 as the distance threshold approaches 0.

## Abstract

Explicit formulas for complexity and unique invariant measure of the period-doubling subshift can be derived from those for the Thue-Morse subshift, obtained by Brlek, De Luca and Varricchio, and Dekking. In this note we give direct proofs based on combinatorial properties of the period-doubling sequence. We also derive explicit formulas for correlation integral and other recurrence characteristics of the period-doubling subshift. As a corollary we obtain that the determinism of this subshift converges to 1 as the distance threshold approaches 0.