Growth of degrees of integrable mappings

We study mappings obtained as s-periodic reductions of the lattice Korteweg-De Vries equation. For small s=(s1,s2) 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 any s1,s2 that are co-prime the growth is ~n^2/(2s1s2), 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 ~n^2(s1+s2)/(2s1s2).