Zeckendorf representations and mixing properties of sequences

TL;DR

This paper explores the use of Zeckendorf representations to analyze the mixing properties of symbolic dynamical systems generated by random substitutions related to Fibonacci, tribonacci, and metallic mean numbers.

Contribution

It introduces a novel approach linking Zeckendorf representations with the mixing behavior of systems from random substitutions, leveraging their numeration schemes.

Findings

Demonstrates mixing properties for systems associated with Fibonacci, tribonacci, and metallic mean numbers.

Establishes a connection between Zeckendorf representations and dynamical system behavior.

Provides a framework for analyzing symbolic systems using number representations.

Abstract

We use generalised Zeckendorf representations of natural numbers to investigate mixing properties of symbolic dynamical systems. The systems we consider consist of bi-infinite sequences associated with so-called random substitutions. We focus on random substitutions associated with the Fibonacci, tribonacci and metallic mean numbers and take advantage of their respective numeration schemes.