Growth of degrees of integrable mappings

We study mappings obtained as s-periodic reductions of the lattice Korteweg-De Vries equation. For small s ∈ N we establish upper bounds on the growth of the degree of the numerator of their iterates. These upper bounds appear to be exact. Moreover, we conjecture that for s1, s2 co-prime the growth is ∼ (2s1s2)−1n2, except when s1 + s2 = 4 where the growth is linear ∼ n. Also, we conjecture the degree of the n-th iterate in projective space to be ∼ (s1 + s2)(2s1s2)−1n2.