A dynamical system in the space of convex quadrangles

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.