Complex Caustics of the Elliptic Billiard

Fierobe, Corentin

doi:10.1007/s40598-020-00152-w

Cited by 4 publications

(3 citation statements)

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“…Indeed, among pairs of real conics, the confocal conics are exactly those that share their isotropic tangents. Confocal conics produce integrable billiards, so it is reasonable to expect ((G5)) to influence the dynamics (see also [Glu14b,Fie21a]). Lemma 6.1.…”

Section: The Dynamical Degree Boundmentioning

confidence: 99%

“…Specializing our arguments to degree d = 2 and appealing to a classification theorem, we obtain a new proof of complete integrability of billiards in quadrics that works over any algebraically closed field of characteristic not equal to 2. For other algebraic approaches to elliptic billiards and Poncelet's porism, see [Dui10,Chapter 11.2] and [Fie21a,GH78]. Corollary 6.11.…”

Section: The Dynamical Degree Boundmentioning

confidence: 99%

“…Billiards in a complex plane curve C 0 ⊆ P 2 C was introduced by Glutsyuk as a correspondence, or meromorphic multi-valued map, on C 0 × P 1 [Glu14a]. (Multivaluedness arises because intersecting a line with an algebraic curve requires taking a root; see Figure 1.) Complex billiards has been used to prove a surprising array of results about real billiards that are not at all obvious from real geometry alone [Fie21a,Fie21b,Glu14b,Glu17,Glu21,Rom14].…”

Section: Introductionmentioning

confidence: 99%

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The algebraic dynamics of the pentagram map

Weinreich

2022

Ergod. Th. Dynam. Sys.

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The pentagram map, introduced by Schwartz [The pentagram map. Exp. Math.1(1) (1992), 71–81], is a dynamical system on the moduli space of polygons in the projective plane. Its real and complex dynamics have been explored in detail. We study the pentagram map over an arbitrary algebraically closed field of characteristic not equal to 2. We prove that the pentagram map on twisted polygons is a discrete integrable system, in the sense of algebraic complete integrability: the pentagram map is birational to a self-map of a family of abelian varieties. This generalizes Soloviev’s proof of complex integrability [F. Soloviev. Integrability of the pentagram map. Duke Math. J.162(15) (2013), 2815–2853]. In the course of the proof, we construct the moduli space of twisted n-gons, derive formulas for the pentagram map, and calculate the Lax representation by characteristic-independent methods.

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Section: The Dynamical Degree Boundmentioning

confidence: 99%

Section: The Dynamical Degree Boundmentioning

confidence: 99%