Circular Pentagons and Real Solutions of Painlevé VI Equations

We study real solutions of a class of Painlevé VI equations. To each such solution we associate a geometric object, a one-parametric family of circular pentagons. We describe an algorithm that permits to compute the numbers of zeros, poles, 1-points and fixed points of the solution on the interval (1,+∞)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${(1,+\infty)}$$\end{document} and their mutual position. The monodromy of the associated linear equation and parameters of the Painlevé VI equation are easily recovered from the family of pentagons.