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A proof of Culter’s theorem on the existence of periodic orbits in polygonal outer billiards
, 2008
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Recent advances in open billiards with some open problems
- In Frontiers in
, 2010
"... Much recent interest has focused on “open ” dynamical systems, in which a classical map or flow is considered only until the trajectory reaches a “hole”, at which the dynamics is no longer considered. Here we consider questions pertaining to the survival probability as a function of time, given an i ..."
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Cited by 10 (2 self)
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Much recent interest has focused on “open ” dynamical systems, in which a classical map or flow is considered only until the trajectory reaches a “hole”, at which the dynamics is no longer considered. Here we consider questions pertaining to the survival probability as a function of time, given an initial measure on phase space. We focus on the case of billiard dynamics, namely that of a point particle moving with constant velocity except for mirror-like reflections at the boundary, and give a number of recent results, physical applications and open problems. A mathematical billiard is a dynamical system in which a point particle moves with constant speed in a straight line in a compact domain D ⊂ Rd with a piecewise smootha boundary ∂D and making mirror-likeb reflections whenever it reaches the boundary. We can assume that the speed and mass
Obtuse Triangular Billiards I: Near the (2,3,6) Triangle to appear
- in Journal of Experimental Mathematics; Preprint; http://www.math.brown.edu/∼res/papers.html
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Right-angled billiards and volumes of the moduli spaces of quadratic differentials
- on CP1, Preprint, 2012. ArXiv, math.DS/1212.1660
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Periodic Billiard Trajectories in Polyhedra
"... Abstract. We consider the billiard map inside a polyhedron. We give a condi-tion for the stability of the periodic trajectories. We apply this result to the case of the tetrahedron. We deduce the existence of an open set of tetrahedra which have a periodic orbit of length four (generalization of Fag ..."
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Abstract. We consider the billiard map inside a polyhedron. We give a condi-tion for the stability of the periodic trajectories. We apply this result to the case of the tetrahedron. We deduce the existence of an open set of tetrahedra which have a periodic orbit of length four (generalization of Fagnano’s orbit for triangles), moreover we can study completely the orbit of points along this coding. 1.
Recent advances in open billiards
"... Much recent interest has focused on “open ” dynamical systems, in which a classical map or flow is considered only until the trajectory reaches a “hole”, at which the dynamics is no longer considered. Here we consider questions pertaining to the survival probability as a function of time, given an i ..."
Abstract
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Much recent interest has focused on “open ” dynamical systems, in which a classical map or flow is considered only until the trajectory reaches a “hole”, at which the dynamics is no longer considered. Here we consider questions pertaining to the survival probability as a function of time, given an initial measure on phase space. We focus on the case of billiard dynamics, namely that of a point particle moving with constant velocity except for mirror-like reflections at the boundary, and give a number of recent results, physical applications and open problems. 1
