Solitons of the midpoint mapping and affine curvature

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