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ERA-MS

We consider billiard ball motion in
a convex domain of a constant curvature surface influenced by the
constant magnetic field. We prove that if the billiard map is
totally integrable then the boundary curve is necessarily a circle.
This result shows that the so-called Hopf rigidity phenomenon which
was recently obtained for classical billiards on constant curvature
surfaces holds true also in the presence of constant magnetic field.

JMD

The main result of this paper is that, in contrast to the 2D case, for convex billiards in
higher dimensions, for every point on the
boundary, and for every $n$, there always exist billiard trajectories
developing conjugate points at the $n$-th collision with the
boundary. We shall explain that this is a consequence of the
following variational property of the billiard orbits in higher
dimension. If a segment of an orbit is locally maximizing, then it
can not pass too close to the boundary. This fact follows from the
second variation formula for the length functional. It turns out
that this formula behaves differently with respect to "longitudinal'' and "transverse'' variations.

DCDS

This paper deals with Hopf type rigidity for convex billiards on
surfaces of constant curvature. I prove that the only convex
billiard without conjugate points on the hyperbolic plane or on the
hemisphere is a circular billiard.

DCDS

Consider a Riemannian metric on two-torus. We prove that the
question of existence of polynomial first integrals leads naturally
to a remarkable system of quasi-linear equations which turns out to
be a Rich system of conservation laws. This reduces the question of
integrability to the question of existence of smooth (quasi-)
periodic solutions for this Rich quasi-linear system.

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