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 ∗ : R d × R d → R d and a vector s ∈ 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 λ = lim n → ∞ g ( n ) 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 λ n ≤ g ( n ) ≤ a ′ n r ′ λ n . While proving the upper bound, we actually also provide another approach in proving the limit λ 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 λ , which are followed by a good estimation of λ when we have the value of g ( n ) for an n large enough.