We study mappings obtained as s-periodic reductions of the lattice Korteweg-de Vries equation. For small, s ∈ℕ 2we 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 s 1, s 2 that are co-prime, the growth is, ~(2s 1s 2) -1 n 2except when, s 1+s 2=4 where the growth is linear ~n. Also, we conjecture the degree of the nth iterate in projective space to be ~(s 1+s 2)(2s 1s 2) -1 n 2.
Funding
This research was funded by the Australian Research Council through the Centre of Excellence for Mathematics and Statistics of Complex Systems. I thank Jarmo Hietarinta for introducing the heuristic approach at the Summer School on Symmetries (SMS) and Integrability of Difference Equations, CRM (2008). The work was further developed during the programme Discrete Integrable Systems at the Isaac Newton Institute (2009) and I am grateful to its hospitality. Thanks to Claude Viallet, Reinout Quispel and Dinh Tran for useful suggestions.
