Parry order of Parry numbers

TL;DR

This paper introduces the Parry order to classify Perron numbers based on their powers being Parry numbers, revealing structural insights and connections to Salem and Pisot numbers.

Contribution

It defines the Parry order, characterizes Perron numbers with infinitely many Parry powers, and provides bounds and examples illustrating the structure of these classes.

Findings

Perron numbers have infinitely many Parry powers iff they are Pisot or Salem.

Explicit upper bounds on Parry order in terms of algebraic properties.

Existence of non-Parry Perron numbers with powers becoming Parry, showing richness of classes.

Abstract

We introduce the \emph{Parry order} $\mathrm{Ord}_P(\beta)$, defined as the largest integer $n$ for which $\beta^n$ is a Parry number. This leads to a natural partition of the set of Perron numbers as follows: \[ \mathcal{P} = \left( \bigcup_{n \geq 0} H_n \right) \cup H_\infty, \] where $H_n$ is the class of Perron numbers with Parry order $n$, and $H_\infty = S \cup T$ consists exactly of all Pisot and Salem numbers. We show that a Perron number has infinitely many Parry powers if and only if it is Pisot or Salem. For every other Perron number, only finitely many powers can be Parry. We give an explicit upper bound on $\mathrm{Ord}_P(\beta)$ in terms of algebraic properties of~$\beta$. We provide explicit examples of non-Parry Perron numbers whose powers become Parry, demonstrating that several $H_n$ are non-empty and structurally rich. We give an infinite family of cubic non-Pisot numbers, all of which have finite Parry order, but where the family has unbounded Parry order. These results establish a new dynamical perspective on Perron numbers, connecting $\beta$-expansion theory with classical questions surrounding Salem numbers and Lehmer-type conjectures.