Growth of bilinear maps

05/19/2020

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by   Vuong Bui, et al.

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For a bilinear map *:ℝ^d×ℝ^d→ℝ^d of nonnegative coefficients and a vector s∈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 λ 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 λ is algebraic.