On the conjugacy class of the Fibonacci dynamical system

TL;DR

This paper characterizes the class of symbolic dynamical systems topologically isomorphic to the Fibonacci system, revealing the existence of infinitely many systems in this class, including some not generated by substitutions.

Contribution

It provides a complete characterization of the Fibonacci conjugacy class and shows the existence of infinitely many non-substitution systems within it.

Findings

Infinitely many primitive substitutions generate systems in the Fibonacci class.

There are infinitely many systems in this class not generated by substitutions.

An example system is obtained by doubling zeros in the Fibonacci word.

Abstract

We characterize the symbolical dynamical systems which are topologically isomorphic to the Fibonacci dynmaical system. We prove that there are infinitely many injective primitive substitutions generating a dynamical system in the Fibonacci conjugacy class. In this class there are infinitely many dynamical systems not generated by a substitution. An example is the system generated by doubling the 0's in the infinite Fibonacci word.