On the Growth Function of \(n\)-Valued Dynamics

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