Solitons of the midpoint mapping and affine curvature

For a polygon x=(x_j)_j∈ℤ in ℝ^n we consider the midpoints polygon (M(x))_j=( x_j+x_j+1) /2. We call a polygon a soliton of the midpoints mapping M if its midpoints polygon is the image of the polygon under an invertible affine map. We show that a large class of these polygons lie on an orbit of a one-parameter subgroup of the affine group acting on ℝ^n. These smooth curves are also characterized as solutions of the differential equation ċ(t)=Bc (t)+d for a matrix B and a vector d . For n=2 these curves are curves of constant generalized-affine curvature k_ga=k_ga(B) depending on B parametrized by generalized-affine arc length unless they are parametrizations of a parabola, an ellipse, or a hyperbola.