Growth of bilinear maps II: Bounds and orders

TL;DR

This paper investigates the growth behavior of iterated applications of bilinear maps on vectors, establishing bounds and asymptotic estimates for the maximum entry value after multiple applications.

Contribution

It provides tight bounds on the growth rate of the maximum entry, including a new approach to estimate the limit growth rate for bilinear maps.

Findings

Established polynomial bounds on $g(n)$ with respect to $ $ and $\\lambda$

Proved the existence of the limit growth rate $\lambda$ for the maximum entry

Provided methods to estimate $\lambda$ from large $n$ data

Abstract

A good range of problems on trees can be described by the following general setting: Given a bilinear map $*:\mathbb R^d\times\mathbb R^d\to\mathbb R^d$ and a vector $s\in\mathbb R^d$, we need to estimate the largest possible absolute value $g(n)$ of an entry over all vectors obtained from applying $n-1$ applications of $*$ to $n$ instances of $s$. When the coefficients of $*$ are nonnegative and the entries of $s$ are positive, the value $g(n)$ is known to follow a growth rate $\lambda=\lim_{n\to\infty} \sqrt[n]{g(n)}$. In this article, we prove that for such $*$ and $s$ there exist nonnegative numbers $r,r'$ and positive numbers $a,a'$ so that for every $n$, \[ a n^{-r}\lambda^n\le g(n)\le a' n^{r'}\lambda^n. \] While proving the upper bound, we actually also provide another approach in proving the limit $\lambda$ itself. The lower bound is proved by showing a certain form of submultiplicativity for $g(n)$. Corollaries include a lower bound and an upper bound for $\lambda$, which are followed by a good estimation of $\lambda$ when we have the value of $g(n)$ for an $n$ large enough.