Solitons of the midpoint mapping and affine curvature (Q2035125)

A polygon \(x=(x_j)_{j\in\mathbb Z}\) in \(\mathbb R^n\) is called a soliton of the midpoints mapping \(M\) given by \[(M(x))_j=\frac{1}{2}x_j+\frac{1}{2}x_{j+1}\] if its midpoints polygon \(M(x)\) is the image of the polygon under an invertible affine map. In this paper, the authors show that a large class of these polygons lie on an orbit of a one-parameter subgroup of the affine group acting on \(\mathbb R^n\). These smooth curves are also characterized as solutions of the differential equation \[\dot{c}(t) = Bc(t) + d\] for a matrix \(B\) and a vector \(d\). For \(n=2\) these curves are curves of constant generalized-affine curvature depending on \(B\) parametrized by generalized-affine arc length unless they are parametrizations of a parabola, an ellipse, or a hyperbola.