Related papers: Trees and flowers on a billiard table
The dynamics of a beam held on a horizontal frame by springs and bouncing off a step is described by a separable two degrees of freedom Hamiltonian system with impacts that respect, point wise, the separability symmetry. The energy in each…
In this paper we study the Birkhoff Normal Form around elliptic periodic points for a variety of dynamical billiards. We give an explicit construction of the Birkhoff transformation and obtain explicit formulas for the first two twist…
The dynamics of three soft interacting particles on a ring is shown to correspond to the motion of one particle inside a soft triangular billiard. The dynamics inside the soft billiard depends only on the {\it masses ratio} between…
We classify the periodic digit strings which arise from periodic billiard orbits on the four convex $n$-gons $\Delta$ which tile $\mathbb{R}^2$ under reflection, answering problem a posed by Baxter and Umble. $\Delta$ is either an…
We study the thermal rectification phenomenon in ``billiard'' systems with interacting particles. This interaction induces a local dynamical response of the billiard to an external thermodynamic gradient. To explain this dynamical effect we…
Much of dynamical algebraic combinatorics focuses on global dynamical systems defined via maps that are compositions of local toggle operators. The second author and Roby studied such maps that result from toggling independent sets of a…
Numerical experiments of the statistical evolution of an ensemble of non-interacting particles in a time-dependent billiard with inelastic collisions, reveals the existence of three statistical regimes for the evolution of the speeds…
We introduce and prove numerous new results about the orbits of the $T$-fractal billiard. Specifically, in Section 3, we give a variety of sufficient conditions for the existence of a sequence of compatible periodic orbits. In Section 4, we…
The flip graph of triangulations has as vertices all triangulations of a convex $n$-gon, and an edge between any two triangulations that differ in exactly one edge. An $r$-rainbow cycle in this graph is a cycle in which every inner edge of…
We define piecewise-linear and birational analogues of the toggle-involutions on order ideals of posets studied by Striker and Williams and use them to define corresponding analogues of rowmotion and promotion that share many of the…
The billiard problem of statistical physics is considered in a new geometric approach with a symmetric phase space. The structure and topological features of typical billiard phase portrait are defined. The connection between geometric,…
We discuss a notion of shuffle for trees which extends the usual notion of a shuffle for two natural numbers. We give several equivalent descriptions, and prove some algebraic and combinatorial properties. In addition, we characterize…
We prove a few simple cases of a random graph statement that would imply the "second" Kahn--Kalai Conjecture. Even these cases turn out to be reasonably challenging, and it is hoped that the ideas introduced here may lead to further…
We study the problem of arithmetic billiards from a new perspective. We first raise a similar problem about reflecting lights inside grids. For the solution to this problem, we will give three proofs. Next, we consider a similar problem in…
In a set of experiments, Couder et. al. demonstrate that an oscillating fluid bed may propagate a bouncing droplet through the guidance of the surface waves. We present a dynamical systems model, in the form of an iterative map, for a…
We study the stability of periodic trajectories of planar inverse magnetic billiards, a dynamical system whose trajectories are straight lines inside a connected planar domain $\Omega$ and circular arcs outside $\Omega$. Explicit examples…
We study a system of an elastic ball moving in the non-relativistic spacetime with a nontrivial causal structure produced by a wormhole-based time machine. For such a system it is possible to formulate a simple model of the so-called…
We study single-flip dynamics in sets of three-dimensional rhombus tilings with fixed polyhedral boundaries. This dynamics is likely to be slowed down by so-called ``cycles'': such structures arise when tilings are encoded via the…
Wire billiard is defined by a smooth embedded closed curve of non-vanishing curvature $k$ in $\mathbb{R}^n$ (a wire). For a class of curves, that we call nice wires, the wire billiard map is area preserving twist map of the cylinder. In…
We identify a collection of periodic billiard orbits in a self-similar Sierpinski carpet billiard table. Based on a refinement of the result of Durand-Cartagena and Tyson regarding nontrivial line segments in a self-similar Sierpinski…