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Working paper

A dynamical system in the space of convex quadrangles

Cornell university preprint series. Arxiv. math. Cornell university, 2021. No. 2106.15557.
Let us consider a family F(alpha,beta,gamma,delta) of convex quadrangles in the plane with given angles  {alpha,beta,gamma,delta} and with the perimeter 2pi. Such quadrangle Q from  F(alpha,beta,gamma,delta) can be considered as a point (x_1,x_2,x_3,x_4) in 4-dimensional space R^4, where {x_1,x_2,x_3,x_4} ---  lengths of edges. Then to F a finite open segment I in R^4 is corresponded. A quadrangle in F, that corresponds to the midpoint of I$is called a balanced quadrangle. Let M be the set of balanced quadrangles. The function f:M -- > M is defined in the following way:  angles of the balanced quadrangle Q', Q'=f(Q), are numerically equal to edges of Q. The map f defines a dynamical system in the space of balanced quadrangles. In this work we study properties of this system.