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 \(*:\mathbb {R}^d\times \mathbb {R}^d\rightarrow \mathbb {R}^d\) 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 \(\lambda =\lim _{n\rightarrow \infty } \root n \of {g(n)}\). 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,

$$\begin{aligned} a n^{-r}\lambda ^n\le g(n)\le a' n^{r'}\lambda ^n. \end{aligned}$$

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.