3 research outputs found
Lens Rigidity in Scattering by Unions of Strictly Convex Bodies in
It was proved in \cite{NS1} that obstacles in that are finite
disjoint unions of strictly convex domains with boundaries are uniquely
determined by the travelling times of billiard trajectories in their exteriors
and also by their so called scattering length spectra. However the case
is not properly covered in \cite{NS1}. In the present paper we give a separate
different proof of the same result in the case
Travelling Times in Scattering by Obstacles in Curved Space
We consider travelling times of billiard trajectories in the exterior of an
obstacle K on a two-dimensional Riemannian manifold M. We prove that given two
obstacles with almost the same travelling times, the generalised geodesic flows
on the non-trapping parts of their respective phase-spaces will have a
time-preserving conjugacy. Moreover, if M has non-positive sectional curvature
we prove that if K and L are two obstacles with strictly convex boundaries and
almost the same travelling times then K and L are identical
Smooth conjugacy classes of 3D Axiom A flows
We show a rigidity result for 3-dimensional contact Axiom A flows: given two
3D contact Axiom A flows whose restrictions
to basic sets
are orbit equivalent, we prove that if periodic orbits in correspondence have
the same length, then the conjugacy is as regular as the flows and respects the
contact structure, extending a previous result due to Feldman-Ornstein [21].
Some of the ideas are reminiscent of the work of Otal [51]. As an application,
we show that the billiard maps of two open dispersing billiards without eclipse
and with the same marked length spectrum are smoothly conjugated.Comment: There was a mistake in Proposition 3.1; it affects the result about
spectral rigidity of open dispersing billiards, which was removed from the
paper. In the present version, we focus on dynamical results. The main
dynamical result has been improved (upgraded regularity). We also discuss the
preservation of symmetries by the conjugac