Billiards in nearly isosceles triangles
W. Patrick Hooper Richard Evan Schwartz
Journal of Modern Dynamics 2009, 3(2): 159-231 doi: 10.3934/jmd.2009.3.159
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
Research announcement: unbounded orbits for outer billiards
Richard Evan Schwartz
Electronic Research Announcements 2007, 14(0): 1-6 doi: 10.3934/era.2007.14.1
keywords: self-similar tilings polygon exchange maps unbounded orbits Piecewise isometries outer billiards Penrose kite
Outer billiards and the pinwheel map
Richard Evan Schwartz
Journal of Modern Dynamics 2011, 5(2): 255-283 doi: 10.3934/jmd.2011.5.255
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
Outer billiards on the Penrose kite: Compactification and renormalization
Richard Evan Schwartz
Journal of Modern Dynamics 2011, 5(3): 473-581 doi: 10.3934/jmd.2011.5.473
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.
Erratum: Billiards in nearly isosceles triangles
W. Patrick Hooper Richard Evan Schwartz
Journal of Modern Dynamics 2014, 8(1): 133-137 doi: 10.3934/jmd.2014.8.133
keywords: trigonometric series Veech triangles periodic orbits unfoldings. isosceles triangles Triangular billiards
Unbounded orbits for outer billiards I
Richard Evan Schwartz
Journal of Modern Dynamics 2007, 1(3): 371-424 doi: 10.3934/jmd.2007.1.371
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

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