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 , + ∞ ) 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.