Circular pentagons and real solutions of Painleve VI equations
  We study real solutions of a class of Painleve VI equations. To each such solution we associate a geometric object, a one-parametric family of circular pentagons. We describe an algorithm which permits to compute the numbers of zeros, poles, 1-points and fixed points of the solution on the interval x>1 and their mutual position. The monodromy of the associated linear equation and parameters of the Painleve VI equation are easily recovered from the family of pentagons. 
Alexandre Eremenko and Andrei Gabrielov
math.CV math-ph math.MP