A dynamical system in the space of convex quadrangles

Yury Kochetkov

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arXiv · 2106.15557

A dynamical system in the space of convex quadrangles

Yury Kochetkov

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Abstract

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 $2\pi$. Such quadrangle $Q\in F(\alpha,\beta,\gamma,\delta)$ can be considered as a point $(x_1,x_2,x_3,x_4)\in\mathbb{R}^4$, where $\{x_1,x_2,x_3,x_4\}$ are lengths of edges. Then to $F$ a finite open segment $I\subset\mathbb{R}^4$ is corresponded. A quadrangle in $F$, that corresponds to the midpoint of $I$ is called a \emph{balanced quadrangle}. Let $M$ be the set of balanced quadrangles. The function $f:M\to M$ 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.

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Yury Kochetkov. 2021-06-29. A dynamical system in the space of convex quadrangles. https://arxiv.org/abs/2106.15557

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