Two dynamical systems in the space of triangles

Abstract

Let $M$ be the space of triangles, defined up to shifts, rotations and dilations. We define two maps $f:M\to M$ and $g:M\to M$. The map $f$ corresponds to a triangle of perimeter $\pi$ the triangle with angles numerically equal to edges of the initial triangle. The map $g$ corresponds to a triangle of perimeter $2\pi$ the triangle with \emph{exterior} angles numerically equal to edges of the initial triangle. For $p\in M$ the sequence $\{p,f(p),f(f(p)),\ldots\}$ converges to the equilateral triangle and the sequence $\{p,g(p),g(g(p)),\ldots\}$ converges to the "degenerate triangle" with angles $(0,0,\pi)$. In Supplement an analogous problem about inscribed-circumscribed quadrangles is discussed.