Block Maps between Primitive Uniform and Pisot Substitutions

TL;DR

This paper establishes a finite classification of block maps between subshifts generated by primitive Pisot or uniform substitutions sharing the same eigenvalue, revealing structural constraints in symbolic dynamics.

Contribution

It introduces a finite set of block maps that exhaustively describe all maps between such subshifts, up to shifts, for the first time.

Findings

Finite set of block maps for primitive Pisot or uniform substitutions

All block maps are characterized by this finite set up to shift

Structural constraints on subshifts with the same eigenvalue

Abstract

In this article, we prove that for all pairs of primitive Pisot or uniform substitutions with the same dominating eigenvalue, there exists a finite set of block maps such that every block map between the corresponding subshifts is an element of this set, up to a shift.