Growth of bilinear maps II: bounds and orders

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