21,295 research outputs found
Rotation sets of billiards with one obstacle
We investigate the rotation sets of billiards on the -dimensional torus
with one small convex obstacle and in the square with one small convex
obstacle. In the first case the displacement function, whose averages we
consider, measures the change of the position of a point in the universal
covering of the torus (that is, in the Euclidean space), in the second case it
measures the rotation around the obstacle. A substantial part of the rotation
set has usual strong properties of rotation sets
A globally stable attractor that is locally unstable everywhere
We construct two examples of invariant manifolds that despite being locally
unstable at every point in the transverse direction are globally stable. Using
numerical simulations we show that these invariant manifolds temporarily repel
nearby trajectories but act as global attractors. We formulate an explanation
for such global stability in terms of the `rate of rotation' of the stable and
unstable eigenvectors spanning the normal subspace associated with each point
of the invariant manifold. We discuss the role of this rate of rotation on the
transitions between the stable and unstable regimes
Polyhedral billiards, eigenfunction concentration and almost periodic control
We study dynamical properties of the billiard flow on convex polyhedra away
from a neighbourhood of the non-smooth part of the boundary, called
``pockets''. We prove there are only finitely many immersed periodic tubes
missing the pockets and moreover establish a new quantitative estimate for the
lengths of such tubes. This extends well-known results in dimension . We
then apply these dynamical results to prove a quantitative Laplace
eigenfunction mass concentration near the pockets of convex polyhedral
billiards. As a technical tool for proving our concentration results on
irrational polyhedra, we establish a control-theoretic estimate on a product
space with an almost-periodic boundary condition. This extends previously known
control estimates for periodic boundary conditions, and seems to be of
independent interest.Comment: 32 pages, a few sections reorganised and a few results adde
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