Chebyshev dynamics on two and three intervals and isomonodromic deformations

Abstract

We present the dynamics of Chebyshev polynomials on d = 2 d=2 and d = 3 d=3 real intervals. This dynamics, for d = 2 d=2 , is associated with isomonodromic deformations and Painlevé VI equations, while for d = 3 d=3 , it is associated with a class of constrained isomonodromic deformations for which we provide the constrained Schlesinger equations. We provide explicit solutions to these equations in terms of differentials on an appropriate family of elliptic curves for d = 2 d=2 and of genus 2 2 curves for d = 3 d=3 . From the point of view of the classical algebraic geometry we present and solve the problem of constrained Jacobi inversion for genus 2 2 curves. We motivate and apply the obtained results to the study of billiards within quadrics in the plane and in 3 3 -dimensional space.