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 c ˙ ( t ) = B c ( 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.