290 research outputs found
Outer Billiards on Kites
Outer billiards is a simple dynamical system based on a convex planar shape.
The Moser-Neumann question, first posed by B.H. Neumann around 1960, asks if
there exists a planar shape for which outer billiards has an unbounded orbit.
The first half of this monograph proves that outer billiards has an unbounded
orbit defined relative to any irrational kite. The second half of the monograph
gives a very sharp description of the set of unbounded orbits, both in terms of
the dynamics and the Hausdorff dimension. The analysis in both halves reveals a
close connection between outer billiards on kites and the modular group, as
well as connections to self-similar tilings, polytope exchange maps,
Diophantine approximation, and odometers.Comment: 296 pages. Essentially, I have added a "second half" to the previous
monograph. Parts I-IV are essentially the same as last posted version. Parts
V-VI have the new materia
Outer Billiards, Arithmetic Graphs, and the Octagon
Outer Billiards is a geometrically inspired dynamical system based on a
convex shape in the plane.
When the shape is a polygon, the system has a combinatorial flavor. In the
polygonal case, there is a natural acceleration of the map, a first return map
to a certain strip in the plane. The arithmetic graph is a geometric encoding
of the symbolic dynamics of this first return map.
In the case of the regular octagon, the case we study, the arithmetic graphs
associated to periodic orbits are polygonal paths in R^8. We are interested in
the asymptotic shapes of these polygonal paths, as the period tends to
infinity. We show that the rescaled limit of essentially any sequence of these
graphs converges to a fractal curve that simultaneously projects one way onto a
variant of the Koch snowflake and another way onto a variant of the Sierpinski
carpet. In a sense, this gives a complete description of the asymptotic
behavior of the symbolic dynamics of the first return map.
What makes all our proofs work is an efficient (and basically well known)
renormalization scheme for the dynamics.Comment: 86 pages, mildly computer-aided proof. My java program
http://www.math.brown.edu/~res/Java/OctoMap2/Main.html illustrates
essentially all the ideas in the paper in an interactive and well-documented
way. This is the second version. The only difference from the first version
is that I simplified the proof of Main Theorem, Statement 2, at the end of
Ch.
A characterization of quasi-rational polygons
The aim of this paper is to study quasi-rational polygons related to the
outer billiard. We compare different notions introduced, and make a synthesis
of those.Comment: 15 pages, 9 figure
Introducing the Plaid Model
We introduce and prove some basic results about a combinatorial model which
produces embedded polygons in the plane. The model is closely related to outer
billiards on kites, and also is related to corner percolation, to Hooper's
Truchet tile system, to self-similar tilings, and to polyhedron exchange
transformations.Comment: 68 pages. This is an more polished version of the original
submission. The connection to polytope exchanges is developed and one of the
speculative sections has been eliminated. Otherwise, it has the same result
Unbounded Orbits for Outer Billiards
Outer billiards is a basic dynamical system, defined relative to a planar
convex shape. This system was introduced in the 1950's by B.H. Neumann and
later popularized in the 1970's by J. Moser. All along, one of the central
questions has been: is there an outer billiards system with an unbounded orbit.
We answer this question by proving that outer billiards defined relative to the
Penrose Kite has an unbounded orbit. The Penrose kite is the quadrilateral that
appears in the famous Penrose tiling. We also analyze some of the finer orbit
structure of outer billiards on the penrose kite. This analysis shows that
there is an uncountable set of unbounded orbits. Our method of proof relates
the problem to self-similar tilings, polygon exchange maps, and arithmetic
dynamics.Comment: 65 pages, computer-aided proof. Auxilliary program, Billiard King,
available from author's website. Latest version is essentially the same as
earlier versions, but with minor improvements and many typos fixe
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