Growth of degrees of integrable mappings

TL;DR

This paper investigates the degree growth of integrable mappings derived from the lattice KdV equation, providing bounds, conjectures, and formulas for their asymptotic behavior based on periodic reductions.

Contribution

It establishes exact upper bounds on degree growth for small periodic reductions and conjectures precise asymptotic growth formulas for general cases.

Findings

Upper bounds on degree growth are exact for small s.

Degree growth is quadratic in n for co-prime s1, s2.

Linear growth occurs when s1+s2=4.

Abstract

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).