Replacement dynamics of binary quadratic forms

TL;DR

This paper studies the dynamics of binary quadratic forms under a process where function outputs replace input entries, classifies periodic vectors, and explores their properties over the rationals.

Contribution

It introduces a stratification of periodic vectors in multivariate polynomial dynamics and classifies rational periodic vectors for quadratic forms up to period 5.

Findings

Classified periodic vectors for quadratic forms up to period 5.

Identified two new types of periodic vectors not from univariate dynamics.

Proved non-existence of certain periodic vectors over rationals.

Abstract

For an $S$-valued function $f$ of $m \geq 1$ variables we consider the dynamical process in which the output $f(\overline{v})$ replaces exactly one entry of the input $\overline{v} \in S^m$ at each step. This can be viewed as a special case of multivariate polynomial semigroup dynamics, and our study focuses on periodic vectors with respect to this process. We define a stratification of periodic vectors according to their type, and characterize types for which the determination of periodic vectors comes down to dynamics of univariate polynomials. We then restrict to the case of a diagonal binary quadratic form $f$ over $\mathbb{Q}$, and classify rational periodic vectors for all types of period up to $5$. This includes two types, of periods $4$ and $5$, which do not arise from the univariate case, and we prove that there are no periodic vectors over the rationals of the single non-univariate type of period $4$.