Replacement dynamics of binary quadratic forms

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 $4$.