Solitons of the midpoint mapping and affine curvature

For a polygon $x=(x_j)_{j\in \mathbb{Z}}$ in $\mathbb{R}^n$ we consider the midpoints polygon $(M(x))_j=\left(x_j+x_{j+1}\right)/2\,.$ 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 $\mathbb{R}^n.$ 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 $k_{ga}=k_{ga}(B)$ depending on $B$ parametrized by generalized-affine arc length unless they are parametrizations of a parabola, an ellipse, or a hyperbola.


Introduction
We consider an infinite polygon (x j ) j∈Z given by its vertices x j ∈ R n in an n-dimensional real vector space R n resp. an n-dimensional affine space A n modelled after R n . For a parameter α ∈ (0, 1) we introduce the polygon M α (x) whose vertices are given by (M α (x)) j := (1 − α)x j + αx j+1 .
For α = 1/2 this defines the midpoints polygon M (x) = M 1/2 (x). On the space P = P(R n ) of polygons in R n this defines a discrete curve shortening process M α : P −→ P, already considered by Darboux [4] in the case of a closed resp. periodic polygon. For a discussion of this elementary geometric construction see Berlekamp et al. [1].
The mapping M α is invariant under the canonical action of the affine group. The affine group Aff(n) in dimension n is the set of affine maps (A, b) : R n −→ R n , x −→ Ax + b.
Here A ∈ Gl(n) is an invertible matrix and b ∈ R n a vector. The translations x −→ x + b determined by a vector b form a subgroup isomorphic to R n . Let α ∈ (0, 1). We call a polygon x j a soliton for the process M α (or affinely invariant under M α ) if there is an affine map (A, b) ∈ Aff(n) such that (1) (M α (x)) j = Ax j + b for all j ∈ Z. In Theorem 1 we describe these solitons explicitely and discuss under which assumptions they lie on the orbit of a one-parameter subgroup of the affine group acting canonically on R n . We call a smooth curve c : R −→ R n a soliton of the mapping M α resp. invariant under the mapping M α if there is for some > 0 a smooth mapping s ∈ (− , ) −→ (A(s), b(s)) ∈ Aff(n) such that for all s ∈ (− , ) and t ∈ R : (2)c s (t) := (1 − α)c(t) + αc(t + s) = A(s)c(t) + b(s) .
Then for some t 0 ∈ R and s ∈ (− , ) the polygon x j = c(js + t 0 ), j ∈ Z is a soliton of M α .
The parabola is an example of a soliton of M = M 1/2 , cf. Figure 1 and Example 1, Case (e). We show in Theorem 2 that the smooth curves invariant under M α coincide with the orbits of a one-parameter subgroup of the affine group Aff(n) acting canonically on R n . For n = 2 we give a characterization of these curves in terms of the general-affine curvature in Section 5. The authors discussed solitons, i.e. curves affinely invariant under the curve shortening in [9]. The solitons of M = M 1/2 form a subclass of the solitons of T, since (T (x)) j = (M 2 (x)) j−1 . Instead of the discrete evolution of polygons one can also investigate the evolution of polygons under a linear flow, cf. Viera and Garcia [11] and [9, sec.4] or a non-linear flow, cf. Glickenstein and Liang [5].
2. The affine group and systems of linear differential equations of first order The affine group Aff(n) is a semidirect product of the general linear group Gl(n) and the group R n of translations. There is a linear representation of the affine group in the general linear group Gl(n+1), cf. [8,Sec.5.1]. We use the following identification (4) A b 0 1 Hence we can identify the image of a vector x ∈ R n under the affine map . Using this identification we can write down the solution of an inhomogeneous system of linear differential equations with constant coefficients using the power series F B (t) which we introduce now: Proposition 1. For a real (n, n)-matrix B ∈ M R (n) we denote by F B (t) ∈ M R (n) the following power series: (a) We obtain for its derivative: The function F B (t) satisfies the following functional equation: resp. for j ∈ Z, j ≥ 1 : The solution c(t) of the inhomogeneous system of linear differential equations with constant coefficients (i.e. B ∈ M R (n, n), d ∈ R n ) and with initial condition v = c(0) is given by: Proof. (a) Equation (6) follows immediately from Equation (5). Then we compute Since F B (0) = 0 Equation (7) follows. And this implies Equation (8).
(b) We can write the solution of the differential equation (8) d dt as follows: which is Equation (9). One could also differentiate Equation (9) and use Equation (6) Remark 1. Equation (2) shows that c(t) is the orbit of the one-parameter subgroup of the affine group Aff(n) acting canonically on R n .

Polygons invariant under
Assume that for α ∈ (0, 1) the value 1 − α is not an eigenvalue of A, i.e. the matrix A α := α −1 (A + (α − 1)1) is invertible. Then the following statements hold: (a) There is a unique polygon x ∈ P(R n ) with x 0 = v which is a soliton for M α resp. affinely invariant under the mapping M α with respect to the affine map (A, b), cf. Equation (1). If b α = α −1 b, then for j > 0 : and for j < 0 : for a vector d α ∈ R n then the polygon x j lies on the smooth curve Proof. (a) By Equation (1) we have for all j ∈ Z. Hence the polygon is given by x 0 = v and the recursion formulae for all j ∈ Z. Then Equation (11) and Equation (12) follow.
. Hence for j > 0 : Note that the matrices B, F B (t), F B (t) −1 commute. With this identity we obtain for j < 0 : (4) we can write

Remark 2. (a) Using the identification Equation
then we obtain from Equation (10): Hence t ∈ R −→ c(t) ∈ R n is the orbit of a one-parameter subgroup of the affine group applied to the vector v.

Smooth curves invariant under M α
For a smooth curve c : R −→ R n and a parameter α ∈ (0, 1) we define the one-parameter familyc s : R −→ R n , s ∈ R by Equation (2). And we call a smooth curve c : R −→ R n a soliton of the mapping M α (resp. affinely invariant under M α ) if there is > 0 and a Then we obtain as an analogue of [9, Thm.1]: Theorem 2. Let c : R −→ R n be a soliton of the mapping M α satisfying Equation (14).
Then the curve c is the unique solution of the differential equatioṅ Hence the curve c(t) is the orbit of a one-parameter subgroup of the affine group, i.e.

Remark 3.
For an affine map (A, b) ∈ Gl(n), b ∈ R n the linear isomorphism A is called the linear part. For n = 2 we discuss the possible normal forms of A ∈ Gl(2) resp. the normal forms of the one-parameter subgroup exp(tB) and of the one-parameter family This will be used in Section 5. ( as well as the one-parameter family A(s) = (1 − α)1 + α exp(Bs) consists of scalings.
The curves c invariant under the process T considered in [9] define a class of curves containing the orbits of the one-parameter subgroups of the affine group. They are solutions of the second order differential equationc = Bc + d which on the other hand can be reduced to a system of first order differential equations, cf. [9, Rem.1].

Curves with constant affine curvature
The orbits of one-parameter subgroups of the affine group Aff(2) acting on R 2 can also be characterized as curves of constant general-affine curvature parametrized proportional to general-affine arc length unless they are parametrizations of a parabola, an ellipse or a hyperbola. This will be discussed in this section. The one-parameter subgroups are determined by an endomorphism B and a vector d. We describe in Proposition 2 how the general-affine curvature can be expressed in terms of the matrix B.
For certain subgroups of the affine group Aff(2) one can introduce a corresponding curvature and arc length. One should be aware that sometimes in the literature the curvature related to the equi-affine subgroup SAff(2) generated by the special linear group SL(2) of linear maps of determinant one and the translations is also called affine curvature. We distinguish in the following between the equi-affine curvature k ea and the general-affine curvature k ga as well as between the equi-affine length parameter s ea and the general-affine length parameter s ga .
We recall the definition of the equi-affine and general-affine curvature of a smooth plane curve c : I −→ R 2 with det(ċ(t)c(t)) = |ċ(t)c(t)| = 0 for all t ∈ I.
By eventually changing the orientation of the curve we can assume |ċ(t)c(t)| > 0 for all t ∈ I. A reference is the book by P. and A.Schirokow [10, §10] or the recent article by Kobayashi and Sasaki [7]. Then s ea (t) := |ċ(t)c(t)| 1/3 dt is called equi-affine arc length.
We denote by t = t(s ea ) the inverse function, thenc(s ea ) = c(t(s ea )) is the parametrization by equi-affine arc length. Thenc (s ea ),c (s ea ) are linearly dependent and the equi-affine curvature k ea (s) is defined byc Assume that c = c(s ea ), s ea ∈ I is a smooth curve parametrized by equi-affine arc length for which the sign = sign(k ea (s)) ∈ {0, ±1} of the equi-affine curvature is constant. If = 0 then the curve is up to an affine transformation a parabola (t, t 2 ). Now assume = 0 and let K ea = |k ea | = k ea . Then the general-affine arc length s ga = s ga (s ea ) is defined by We call a curve c = c(t) parametrized proportional to general-affine arc length if t = λ 1 s ga + λ 2 for λ 1 , λ 2 ∈ R with λ 1 = 0. The general-affine curvature k ga = k ga (s) is defined by If the general-affine curvature k ga (up to sign) and the sign is given with respect to (b) If tr(B) = 0 then we can choose a parametrization by equi-affine arc length s ea such that the equi-affine curvature k ea is given by: If k(B) = 0 the curve has vanishing equi-affine curvature and is a parametrization of a parabola, cf. the Remark 4. If k(B) = 0 then the general-affine curvature is defined and constant: (23) k ga (s ea ) = −2K(B).
Up to an additive constant the general-affine arc length parameter s ga is given by: Hence the curve c(t) is parametrized proportional to general-affine arc length.  Hence the equi-affine arc length parametrization of c is given bỹ Then we can express the derivatives:  If λ = −1 we obtain for the equi-affine curvature with respect to a equi-affine parametrization s ea from Equation (22): For λ = 1/2, 2 we obtain a parametrization of a parabola with vanishing equi-affine curvature, cf. Remark 4. Now we assume λ = 1/2, 2. Hence = 1 if and only if 1/2 < λ < 2. The affine curvature k ga is constant:  exp(at) (cos(bt), sin(bt)). For a = 0, this is a circle with k ea = 1. Now we assume a = 0 : Then = sign(9 det(B) − 2tr 2 (B)) = sign(a 2 + 9b 2 ) = 1 and we obtain for the general-affine curvature k ga = −4|a|/ √ a 2 + 9b 2 = −4|a|/ √ 9 − 8a 2 , i.e. k ga ∈ (−4, 0). The corresponding one-parameter subgroup B(t) as well as the oneparameter family A(s) consist of similarities. For the affine curve shortening flow the parabola is a translational soliton. Therefore it is also called the affine analogue of the grim reaper, cf. [3, p.192]. For the curve shortening process T defined by Equation (3) the parabola is also a translational soliton, cf. [9,Sec.5,Case (5)].
Note that the parabola occurs twice, in Case (a) it occurs with the parametrization c(t) = (exp(t), exp(2t)), in Case (e) it occurs with a parametrization proportional to equiaffine arc length. Summarizing we obtain from Theorem 2 and Theorem 3 together with Proposition 2 resp. Example 1 the following Theorem 4. Let c : R −→ R 2 be a smooth curve for whichċ(0),c(0) are linearly independent. Then c is a soliton of the mappings M α , α ∈ (0, 1), in particular of the midpoints mapping M = M 1/2 , if it is a curve of constant equi-affine curvature parametrized proportional to equi-affine arc length, or a parabola with the parametrization c(t) = (exp(t), exp(2t)) up to an affine transformation, or if it is a curve of constant general-affine curvature parametrized proportional to general-affine arc length.