Geometry of Refractions and Reflections Through a Biperiodic Medium

Glendinning, Paul

doi:10.1137/15m1014127

Cited by 8 publications

(5 citation statements)

References 14 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Glendinning [6] studies a system that is similar to ours: a "chessboard" tiling where the two colors of tiles have different positive indices of refraction, in particular where the refraction coefficient is greater than 1. This is similar to our setup, except that the angle at a boundary changes, which leads to different behavior than in our system.…”

Section: Theorem 22 (Engelman and Kimball) (A)mentioning

confidence: 95%

Negative refraction and tiling billiards

Davis

DiPietro

Rustad

et al. 2018

Advances in Geometry

View full text Add to dashboard Cite

show abstract

Section: Theorem 22 (Engelman and Kimball) (A)mentioning

confidence: 95%

Negative refraction and tiling billiards

Davis

DiPietro

Rustad

et al. 2018

Advances in Geometry

View full text Add to dashboard Cite

show abstract

“…There is a wide literature on Birkhoff billiards, with recent relevant advances (see the book [33] and papers [22,23,20,4]), including some cases of composite billiard with reflections and refractions [3], also in the case of a periodic inhomogenous lattice [17]. Special mention should be paid to the work on magnetic billiards, where the trajectories of a charged particle in this setting are straight lines concatenated with circular arcs of a given Larmor radius [15,16].…”

mentioning

confidence: 99%

On some refraction billiards

Blasi¹,

Terracini²

2023

DCDS

View full text Add to dashboard Cite

<p style='text-indent:20px;'>The aim of this work is to continue the analysis, started in [<xref ref-type="bibr" rid="b10">10</xref>], of the dynamics of a point-mass particle <inline-formula><tex-math id="M1">\begin{document}$ P $\end{document}</tex-math></inline-formula> moving in a galaxy with an harmonic biaxial core, in whose center sits a Keplerian attractive center (e.g. a Black Hole). Accordingly, the plane <inline-formula><tex-math id="M2">\begin{document}$ \mathbb{R}^2 $\end{document}</tex-math></inline-formula> is divided into two complementary domains, depending on whether the gravitational effects of the galaxy's mass distribution or of the Black Hole prevail. Thus, solutions alternate arcs of Keplerian hyperbolæ with harmonic ellipses; at the interface, the trajectory is refracted according to Snell's law. The model was introduced in [<xref ref-type="bibr" rid="b11">11</xref>], in view of applications to astrodynamics.</p><p style='text-indent:20px;'>In this paper we address the general issue of periodic and quasi-periodic orbits and associated caustics when the domain is a perturbation of the circle, taking advantage of KAM and Aubry-Mather theories.</p>

show abstract

“…A tiling billiard is a model of movement of light in a heterogeneous medium that is constructed as a union of homogeneous pieces, see [16] and [18] for the first mathematical approaches of the subject and definitions. The defintion of a tiling billiard is the following.…”

Section: Introduction Motivation and Overview Of Resultsmentioning

confidence: 99%

“…For all of the maps T = F 2 with F ∈ CET 3 τ their SAF invariant is zero. 18 More generally, a square of any fully flipped interval exchange transformation has a zero SAF invariant. This statement has already been proven in [Proposition 18 in [23]].…”

Section: The Lengths Of These Intervals Verify |Imentioning

confidence: 99%

Trees and flowers on a billiard table

Paris-Romaskevich¹

2019

Preprint

View full text Add to dashboard Cite

In this work we completely describe the dynamics of triangle tiling billiards. In the first part of this work, we propose a geometric approach of dynamics by introducing natural foliations associated to it. In the second part, we exploit the relationship between triangle tiling billiards and a family of fully flipped 3-interval exchange transformations on the circle. We give a combinatorial approach of dynamics via renormalization. By uniting the two approaches, we prove several conjectures on the dynamics of triangle tiling billiards. First, we prove the Tree Conjecture and the 4n+2 Conjecture, both concerning the symbolic dynamics of periodic trajectories, and both stated by Baird-Smith, Davis, Fromm and Iyer. Second, we study a family of exceptional trajectories which are closely related to the orbits of minimal Arnoux-Rauzy maps. We prove that all of these exceptional trajectories pass by all tiles, which confirms our own conjecture with P. Hubert on their non-linear escape. Moreover, we use tiling billiards to prove the convergence, up to rescaling, of arithmetic orbits of the Arnoux-Yoccoz map to the Rauzy fractal, conjectured by Hooper and Weiss. All of these conjectures have been stated in print in the last three years.

show abstract

Geometry of Refractions and Reflections Through a Biperiodic Medium

Cited by 8 publications

References 14 publications

Negative refraction and tiling billiards

Negative refraction and tiling billiards

On some refraction billiards

Trees and flowers on a billiard table

Contact Info

Product

Resources

About