26 research outputs found
On recurrence and ergodicity for geodesic flows on noncompact periodic polygonal surfaces
We study the recurrence and ergodicity for the billiard on noncompact
polygonal surfaces with a free, cocompact action of or . In the
-periodic case, we establish criteria for recurrence. In the more difficult
-periodic case, we establish some general results. For a particular
family of -periodic polygonal surfaces, known in the physics literature
as the wind-tree model, assuming certain restrictions of geometric nature, we
obtain the ergodic decomposition of directional billiard dynamics for a dense,
countable set of directions. This is a consequence of our results on the
ergodicity of \ZZ-valued cocycles over irrational rotations.Comment: 48 pages, 12 figure
Hyperbolic outer billiards : a first example
We present the first example of a hyperbolic outer billiard. More precisely
we construct a one parameter family of examples which in some sense correspond
to the Bunimovich billiards.Comment: 11 pages, 8 figures, to appear in Nonlinearit
The Simplest Piston Problem II: Inelastic Collisions
We study the dynamics of three particles in a finite interval, in which two
light particles are separated by a heavy ``piston'', with elastic collisions
between particles but inelastic collisions between the light particles and the
interval ends. A symmetry breaking occurs in which the piston migrates near one
end of the interval and performs small-amplitude periodic oscillations on a
logarithmic time scale. The properties of this dissipative limit cycle can be
understood simply in terms of an effective restitution coefficient picture.
Many dynamical features of the three-particle system closely resemble those of
the many-body inelastic piston problem.Comment: 8 pages, 7 figures, 2-column revtex4 forma
Addendum to: Capillary floating and the billiard ball problem
We compare the results of our earlier paper on the floating in neutral
equilibrium at arbitrary orientation in the sense of Finn-Young with the
literature on its counterpart in the sense of Archimedes. We add a few remarks
of personal and social-historical character.Comment: This is an addendum to my article Capillary floating and the billiard
ball problem, Journal of Mathematical Fluid Mechanics 14 (2012), 363 -- 38
Escape orbits and Ergodicity in Infinite Step Billiards
In a previous paper we defined a class of non-compact polygonal billiards,
the infinite step billiards: to a given decreasing sequence of non-negative
numbers , there corresponds a table \Bi := \bigcup_{n\in\N} [n,n+1]
\times [0,p_{n}].
In this article, first we generalize the main result of the previous paper to
a wider class of examples. That is, a.s. there is a unique escape orbit which
belongs to the alpha and omega-limit of every other trajectory. Then, following
a recent work of Troubetzkoy, we prove that generically these systems are
ergodic for almost all initial velocities, and the entropy with respect to a
wide class of ergodic measures is zero.Comment: 27 pages, 8 figure