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### Open Access Journals

ERA-MS

The pentagram map is a projectively natural iteration defined on polygons, and also on
a generalized notion of a polygon which we call

*twisted polygons*. In this note we describe our recent work on the pentagram map, in which we find a Poisson structure on the space of twisted polygons and show that the pentagram map relative to this Poisson structure is completely integrable in the sense of Arnold-Liouville. For certain families of twisted polygons, such as those we call*universally convex*, we translate the integrability into a statement about the quasi-periodic motion of the pentagram-map orbits. We also explain how the continuous limit of the pentagram map is the classical Boussinesq equation, a completely integrable P.D.E.
keywords:
polygons
,
Poisson structre
,
pentagram
,
projective geometry
,
integrability
,
monodromy
,
Bouissinesq equation.
,
iteration

ERA-MS

JMD

In this paper we establish an equivalence between an outer billiards system
based on a convex polygon $P$ and an auxiliary system, which we call the
pinwheel map, that is based on $P$ in a different way. The pinwheel map is
akin to a first-return map of the outer billiards map. The virtue of our
result is that most of the main questions about outer billiards can be
formulated in terms of the pinwheel map, and the pinwheel map is simpler
and seems more amenable to fruitful analysis.

JMD

We give a fairly complete analysis of outer billiards on the Penrose
kite. Our analysis reveals that this $2$-dimensional dynamical system
has a $3$-dimensional compactification, a certain polyhedron exchange
map defined on the $3$-torus, and that this $3$-dimensional system
admits a renormalization scheme. The two features allow us to make
sharp statements concerning the distribution, large- and fine-scale
geometry, and hidden algebraic symmetry, of the orbits. One concrete
result is that the union of the unbounded orbits has Hausdorff
dimension $1$. We establish many of the results with computer-aided
proofs that involve only integer arithmetic.

keywords:
renormalization
,
polytope exchange
,
Dynamics
,
Penrose kite
,
compactification
,
outer billiards
,
piecewise translation.

JMD

N/A

keywords:
trigonometric series
,
Veech triangles
,
periodic orbits
,
unfoldings.
,
isosceles triangles
,
Triangular billiards

JMD

The question of B.H. Neumann, which dates back to the 1950s, asks if there exists an outer billiards system with an unbounded orbit. We prove that outer billiards for the Penrose kite, the convex quadrilateral from the Penrose tiling, has an unbounded orbit. We also analyze some finer properties of the orbit structure, and in particular produce an uncountable family of unbounded orbits. Our methods relate outer billiards on the Penrose kite to polygon exchange maps, arithmetic dynamics, and self-similar tilings.

keywords:
arithmetic graph.
,
Penrose kite
,
unbounded orbits
,
polygon
exchange
,
outer billiards
,
dual billiards

JMD

We prove that any sufficiently small perturbation
of an isosceles triangle has a periodic
billiard path. Our proof involves the analysis
of certain infinite families of Fourier series
that arise in connection with triangular
billiards, and reveals some self-similarity
phenomena in irrational triangular
billiards. Our analysis illustrates the surprising
fact that billiards on a triangle

*near*a Veech triangle is extremely complicated even though billiards*on*a Veech triangle is well understood.
keywords:
trigonometric series
,
Veech triangles
,
unfoldings.
,
isoceles triangles
,
periodic orbits
,
triangular billiards

## Year of publication

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