Growth of bilinear maps

TL;DR

This paper investigates the asymptotic growth rate of the largest entry in repeated applications of a nonnegative bilinear map, establishing the existence of a limit and exploring its algebraic properties.

Contribution

It proves the existence of the growth rate limit using a new structural concept called linear pattern and raises questions about its algebraic nature.

Findings

Existence of the growth rate limit $\\lambda$ proven.

Introduction of linear pattern structure for analysis.

Open question on the algebraic nature of $\lambda$.

Abstract

For a bilinear map $*:\mathbb R^d\times \mathbb R^d\to \mathbb R^d$ of nonnegative coefficients and a vector $s\in \mathbb R^d$ of positive entries, among an exponentially number of ways combining $n$ instances of $s$ using $n-1$ applications of $*$ for a given $n$, we are interested in the largest entry over all the resulting vectors. An asymptotic behavior is that the $n$-th root of this largest entry converges to a growth rate $\lambda$ when $n$ tends to infinity. In this paper, we prove the existence of this limit by a special structure called linear pattern. We also pose a question on the possibility of a relation between the structure and whether $\lambda$ is algebraic.