ERA-MS
Quasiperiodic motion for the pentagram map
Valentin Ovsienko Richard Schwartz Serge Tabachnikov
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
Research announcement: unbounded orbits for outer billiards
Richard Evan Schwartz
keywords: self-similar tilings polygon exchange maps unbounded orbits Piecewise isometries outer billiards Penrose kite
JMD
Outer billiards and the pinwheel map
Richard Evan Schwartz
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.
keywords: piecewise translations pinwheel map. Outer billiards
JMD
Outer billiards on the Penrose kite: Compactification and renormalization
Richard Evan Schwartz
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
Erratum: Billiards in nearly isosceles triangles
W. Patrick Hooper Richard Evan Schwartz
N/A
keywords: trigonometric series Veech triangles periodic orbits unfoldings. isosceles triangles Triangular billiards
JMD
Unbounded orbits for outer billiards I
Richard Evan Schwartz
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
Billiards in nearly isosceles triangles
W. Patrick Hooper Richard Evan Schwartz
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

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