The Pisot Conjecture for $\beta$-substitutions

TL;DR

This paper proves the Pisot Conjecture for beta-substitutions, showing that the associated tiling dynamical system has pure discrete spectrum for Pisot numbers, with implications for arithmetical coding and finitary properties.

Contribution

It establishes the Pisot Conjecture for beta-substitutions, demonstrating pure discrete spectrum and deriving new properties for Pisot numbers.

Findings

Proves pure discrete spectrum for beta-substitutions with Pisot numbers

Shows arithmetical coding is almost everywhere one-to-one

Establishes that every Pisot number is weakly finitary

Abstract

We prove the Pisot Conjecture for beta-substitutions: If beta is a Pisot number, the tiling dynamical system associated with the beta-substitution has pure discrete spectrum. As corollaries: (1) arithmetical coding of the hyperbolic solenoidal automorphism associated with the companion matrix of the minimal polynomial of any Pisot number is almost everywhere one-to-one; and (2) every Pisot number is weakly finitary.