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## On infinitely many foliations by caustics in strictly convex non-closed billiards

Working papers by Cornell University.
Cornell University
,
2021.
No. 2104.01362.

Reflection in strictly convex bounded planar billiard acts on the space of oriented lines and preserves a standard area form. A caustic is a curve C whose tangent lines are reflected by the billiard to lines tangent to C. The famous Birkhoff conjecture states that the only strictly convex billiards with a foliation by closed caustics near the boundary are ellipses. By Lazutkin's theorem, there always exists a Cantor family of closed caustics approaching the boundary. In the present paper we deal with an open billiard, whose boundary is a strictly convex embedded (non-closed) curve γ. We prove that there exists a domain Uγ adjacent to γ from the convex side and a $C^\infty$-smooth foliation of
U\cup γ whose leaves are γ and (non-closed) caustics of the billiard. This generalizes a previous result by R.Melrose on existence of a germ of foliation as above. We show that there exist a continuum of above foliations by caustics whose germs at each point in γ are pairwise different. We prove a more general version of this statement for γ being an (immersed) arc. It also applies to a billiard bounded by a closed strictly convex curve γ and yields infinitely many
"immersed" foliations by immersed caustics. For the proof of the above results, we state and prove their analogue
for a special class of area-preserving maps generalizing billiard reflections: the so-called $C^{\infty}$-lifted strongly billiard-like maps. We also prove a series of results on conjugacy of billiard maps near the boundary for open curves of the above type.

Glutsyuk A., Journal of Fixed Point Theory and Applications 2022 Vol. 24 No. 2 Article 35

For a given closed convex planar curve γ with smooth boundary and a given p>0, the string construction yields a family of curves Γp for which γ is a caustic. The action of the reflection Tp on the tangent lines to γ≃S1 induces its action on the tangency points: a circle diffeomorphism Tp:γ→γ. We say ...

Added: November 4, 2021

Glutsyuk A., On commuting billiards in higher dimensional spaces of constant curvature / Cornell University. Series "Working papers by Cornell University". 2018.

We consider two nested billiards in Rd, d >2, with C2-smooth
strictly convex boundaries. We prove that if the corresponding actions
by reflections on the space of oriented lines commute, then the billiards
are confocal ellipsoids. This together with the previous analogous result of the author in two dimensions solves completely the Commuting
Billiard Conjecture due to Sergei Tabachnikov. ...

Added: August 21, 2018

Glutsyuk A., Pacific Journal of Mathematics 2020 Vol. 305 No. 2 P. 577-595

We consider two nested billiards in ℝd, d≥3, with C2-smooth strictly convex boundaries. We prove that if the corresponding actions by reflections on the space of oriented lines commute, then the billiards are confocal ellipsoids. This together with the previous analogous result of the author in two dimensions solves completely the Commuting Billiard Conjecture due ...

Added: October 12, 2019

A.V.Zabrodin, Zotov A. V., Liashyk A. et al., Theoretical and Mathematical Physics 2017 Vol. 192 No. 2 P. 1141-1153

We discuss the correspondence between models solved by the Bethe ansatz and classical integrable systems of the Calogero type. We illustrate the correspondence by the simplest example of the inhomogeneous asymmetric six-vertex model parameterized by trigonometric(hyperbolic) functions. ...

Added: October 26, 2017

Glutsyuk A., Izmestiev I., Tabachnikov S., Israel Journal of Mathematics 2021 Vol. 241 P. 693-719

By a classical result of Darboux, a foliation of a Riemannian surface has the Graves property (also known as the strong evolution property) if and only if the foliation comes from a Liouville net. A similar result of Blaschke says that a pair of orthogonal foliations has the Ivory property if and only if they ...

Added: October 19, 2020

Hutsalyuk A., Liashyk A, Pakuliak S. Z. et al., Nuclear Physics B 2017 Vol. 923 P. 277-311

We study scalar products of Bethe vectors in the models solvable by the nested algebraic Bethe ansatz and described by superalgebra. Using coproduct properties of the Bethe vectors we obtain a sum formula for their scalar products. This formula describes the scalar product in terms of a sum over partitions of Bethe parameters. We also obtain recursions for ...

Added: October 26, 2017

Glutsyuk A., On curves with Poritsky property / Cornell University. Series "Working papers by Cornell University". 2019.

For a given closed convex planar curve γ with smooth boundary and a given p>0, the string construction yields a family of curves Γp for which γ is a caustic. The action of the reflection Tp on the tangent lines to γ≃S1 induces its action on the tangency points: a circle diffeomorphism p:γ→γ. We say ...

Added: November 12, 2019

Gontsov R. R., V.A. Poberezhnyi, Helminck G. F., Russian Mathematical Surveys 2011 Vol. 66 No. 1 P. 63-105

This article concerns deformations of meromorphic linear differential systems. Problems relating to their existence and classification are reviewed, and the global and local behaviour of solutions to deformation equations in a neighbourhood of their singular set is analysed. Certain classical results established for isomonodromic deformations of Fuchsian systems are generalized to the case of integrable ...

Added: September 27, 2013

Glutsyuk A., Shustin E., Mathematische Annalen 2018 Vol. 372 P. 1481-1501

We show that every polynomially integrable planar outer convex billiard is elliptic. We also
prove an extension of this statement to non-convex billiards. ...

Added: June 29, 2018

Glutsyuk A., Density of thin film billiard reflection pseudogroup in Hamiltonian symplectomorphism pseudogroup / Cornell University. Series arXiv "math". 2021.

Reflections from hypersurfaces act by symplectomorphisms on the space of oriented lines with respect to the canonical symplectic form. We consider an arbitrary infinitely-smooth hypersurface in Euclidean space that is either a global strictly convex closed hypersurface, or a germ of hypersurface. We deal with the pseudogroup generated by compositional ratios of reflections from it ...

Added: October 19, 2020

Skripchenko A., Troubetzkoy S., Annales de l'Institut Fourier 2015 Vol. 65 No. 5 P. 1881-1896

We study the billiard on a square billiard table with a one-sided vertical mirror.
We associate trajectories of these billiards with double rotations and study orbit behavior and questions of complexit ...

Added: March 2, 2016

Poberezhny V. A., On isomonodromic deformations and integrability concerning linear systems of differential equations / ИТЭФ. Series "Препринты ИТЭФ". 2013. No. 18/13.

We review the modern theory of isomonodromic deformations, considering linear systems of
differential equations. On that background we illustrate the natural relations between
such phenomena as integrability, isomonodromy and Painlev\'{e} property. The recent
advances in the theory of isomonodromic deformations we present show perfect agreement to
that approach. ...

Added: March 31, 2014

V. A. Poberezhny, Journal of Mathematical Sciences 2013 Vol. 195 No. 4 P. 533-540

We consider systems of linear differential equations discussing some classical and modern results in the Riemann problem, isomonodromic deformations, and other related topics. Against this background, we illustrate the relations between such phenomena as the integrability, the isomonodromy, and the Painlevé property. The recent advances in the theory of isomonodromic deformations presented show perfect agreement ...

Added: February 14, 2014

Hutsalyuk A., Liashyk A., Pakuliak S. Z. et al., Nuclear Physics B 2018 Vol. 926 P. 256-278

We study quantum integrable models solvable by the nested algebraic Bethe ansatz and possessing gl(m|n)-invariant R-matrix. We compute the norm of the Hamiltonian eigenstates. Using the notion of a generalized model we show that the square of the norm obeys a number of properties that uniquely fix it. We also show that a Jacobian of the system ...

Added: September 13, 2018

Hutsalyuk A., Liashyk A., Pakuliak S. Z. et al., Russian Mathematical Surveys 2017 Vol. 72 No. 1 P. 33-99

Bethe vectors are found for quantum integrable models associated with the supersymmetric Yangians in terms of the current generators of the Yangian double . The method of projections onto intersections of different types of Borel subalgebras of this infinite-dimensional algebra is used to construct the Bethe vectors. Calculation of these projections makes it possible to express the ...

Added: October 26, 2017

Glutsyuk A., Journal of the European Mathematical Society 2021 Vol. 23 No. 3 P. 995-1049

We present a solution of the algebraic version of Birkhoff Conjecture on integrable billiards. Namely we show that every polynomially integrable real bounded convex planar billiard with smooth boundary is an ellipse. We extend this result to billiards with piecewise-smooth and not necessarily convex boundary on arbitrary two-dimensional surface of constant curvature: plane, sphere, Lobachevsky ...

Added: October 12, 2019

Glutsyuk A., Doklady Mathematics 2018 Vol. 98 No. 6 P. 382-385

The algebraic version of the Birkhoff conjecture is solved completely for billiards with a piecewise
$C^2$-smooth boundary on surfaces of constant curvature: Euclidean plane, sphere, and Lobachevsky plane. Namely, we obtain a complete classification of billiards for which the billiard geodesic flow has a nontrivial
first integral depending polynomially on the velocity. According to this classification, every ...

Added: August 25, 2018

Glutsyuk A., Israel Journal of Mathematics 2021

Reflections from hypersurfaces act by symplectomorphisms on the space of oriented lines with respect to the canonical symplectic form. We consider an arbitrary infinitely-smooth hypersurface in Euclidean space that is either a global strictly convex closed hypersurface, or a germ of hypersurface. We deal with the pseudogroup generated by compositional ratios of reflections from it ...

Added: November 4, 2021

Lyashik A., Pakuliak S. Z., Ragoucy E. et al., Journal of Statistical Mechanics: Theory and Experiment 2019 Vol. 2019 No. 4 P. 1-23

We consider quantum integrable models solvable by the nested algebraic Bethe ansatz and possessing gl(N)-invariant R-matrix. We study two types of Bethe vectors. The first type corresponds to the original monodromy matrix. The second type is associated to a monodromy matrix closely related to the inverse of the monodromy matrix. We show that these two types of ...

Added: June 6, 2019

Hutsalyuk A., Liashyk A., Pakuliak S. Z. et al., SciPost Physic (Нидерланды) 2018 Vol. 4 No. 006 P. 1-30

We obtain recursion formulas for the Bethe vectors of models with periodic boundary conditions solvable by the nested algebraic Bethe ansatz and based on the quantum affine algebra U_q(gl_n). We also present a sum formula for their scalar products. This formula describes the scalar product in terms of a sum over partitions of the Bethe parameters, ...

Added: September 13, 2018

Glutsyuk A., On rationally integrable planar dual and projective billiards / Cornell University. Series arXiv "math". 2021.

A caustic of a strictly convex planar bounded billiard is a smooth curve whose tangent lines are reflected from the billiard boundary to its tangent lines. The famous Birkhoff Conjecture states that if the billiard boundary has an inner neighborhood foliated by closed caustics, then the billiard is an ellipse. It was studied by many ...

Added: February 3, 2022

Liashyk A., Slavnov N. A., Journal of High Energy Physics 2018 Vol. 06 No. 018 P. 1-31

We consider quantum integrable models solvable by the nested algebraic Bethe ansatz and possessing gl_3-invariant R-matrix. We study a new recently proposed approach to construct on-shell Bethe vectors of these models. We prove that the vectors constructed by this method are semi-on-shell Bethe vectors for arbitrary values of Bethe parameters. They thus do become on-shell vectors provided ...

Added: September 13, 2018

Vassiliev V., Moscow Mathematical Journal 2017 Vol. 17 No. 4 P. 825-836

The local multiplicities of the caustics, the Maxwell sets, and the complex Stokes' sets in the spaces of versal deformations of Pham singularities (that is, of germs of holomorphic functions C^n --> C^1 which are expressed by the sums of degrees of the coordinate functions) are calculated ...

Added: December 27, 2017

Skripchenko A., Troubetzkoy S., Entropy and Complexity of Polygonal Billiards with Spy mirrors / Cornell University. Series math "arxiv.org". 2015. No. 1501.04584.

We prove that a polygonal billiard with one-sided mirrors has zero
topological entropy. In certain cases we show sub exponential and for other
polynomial estimates on the complexity. ...

Added: January 26, 2015