A dynamical system in the space of convex quadrangles

Yu. Yu. Kochetkov

Fundamentalʹnaâ i prikladnaâ matematika, Tome 25 (2024) no. 2, pp. 183-188

Résumé

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 a 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 $F$ there corresponds a finite open segment $I\subset\mathbb{R}^4$. 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 $f\colon M\to M$ 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.

Bibliographie

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[2] Cantarella J., Needham T., Shonkwiler C., “Random triangles and polygons in the plane”, Amer. Math. Monthly, 126:2 (2019), 113–134 | DOI | MR | Zbl

[3] Kochetkov Yu., Two dynamical systems in the space of triangles, 2021, arXiv: 2101.03734 | Zbl