On Growth Function of $n$-Valued Dynamics

This paper answers the question of V. M. Buchstaber on growth function in case of certain $n$-valued group. This question is in close relation to specific discrete integrable systems. In this paper we find a specific formula for growth function given the case of prime $n$. We also prove a polynomial asymptotic estimate of the growth function in general case. At the end we pose new conjectures and questions regarding growth functions.