Solitons of the midpoint mapping and affine curvature

2020/07/28 by Rademacher, Christine, Rademacher, Hans-Bert
#51M04 (15A16 #53A15) #Differential Geometry (math.DG) #FOS: Mathematics #Metric Geometry (math.MG)

paper · doi:10.48550/arxiv.2007.14067

Abstract

For a polygon x=(x_j)_{j∈ ℤ} in ℝⁿ 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 ℝⁿ. These smooth curves are also characterized as solutions of the differential equation 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 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.

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