Alexey Glutsyuk

On polynomially integrable planar outer billiards and curves with symmetry property

We show that every polynomially integrable planar outer convex billiard is elliptic. We also prove an extension of this statement to non-convex billiards.

On simultaneous uniformization and local nonuniformizability

Abstract We prove existence of a one-dimensional holomorphic foliation (with isolated irremovable singularities) tangent to a rational vector field on appropriate affine algebraic surface of dimension 2 such that the family of leaves intersecting arbitrary given cross-section does not admit a uniformization holomorphic in the parameter by a family of simply connected domains in C . We show that such a foliation can be chosen transversally affine, having a Liouvillian first integral, with dense and hyperbolic leaves and an attracting cycle. This extends the authors result [4] giving a negative answer to Ilyashenkos simultaneous uniformization conjecture and answers negatively to the local version of this conjecture recently proposed by Shcherbakov. To cite this article: A. Glutsyuk, C. R. Acad. Sci. Paris, Ser. I 334 (2002) 489–494.

On determinants of modified Bessel functions and entire solutions of double confluent Heun equations

We investigate the question on existence of entire solutions of well-known linear differential equations that are linearizations of nonlinear equations modeling the Josephson effect in superconductivity. We consider the modified Bessel functions

On monodromy eigenfunctions of Heun equations and boundaries of phase-lock areas in a model of overdamped Josephson effect

I_j(x)

A first integrability result for Miquel dynamics

of the first kind, which are Laurent series coefficients of the analytic function family

On Constrictions of Phase-Lock Areas in Model of Overdamped Josephson Effect and Transition Matrix of the Double-Confluent Heun Equation

e^{\frac x2(z+\frac 1z)}

On families of differential equations on two-torus with all phase-lock areas

. For every

NO PLANAR BILLIARD POSSESSES AN OPEN SET OF QUADRILATERAL TRAJECTORIES

l\geq1

On 4-Reflective Complex Analytic Planar Billiards

we study the family parametrized by

On polynomially integrable Birkhoff billiards on surfaces of constant curvature

k, n\in\mathbb Z^l