Periodic Reporting for period 2 - SPERIG (Spectral rigidity and integrability for billiards and geodesic flows)
Reporting period: 2022-09-01 to 2024-02-29
The central topic of the proposal is a classical question: Can you hear the shape of a drum? This question is over a century old problem that attracted a lot of attention in the mathematics and physics community. In a related direction we worked on the so-called Birkhoff Conjecture stating that the only integrable billiard is an ellipse. Areas of mathematics that were developed to solve “Can you hear the shape of a drum?”. Any progress on this classical problem should find application in various applications including spectral theory, dynamical systems, acoustic design. During the starting period of the proposal we have been working on both questions at depth. In the direction of Birkhoff Conjecture we established it in two different directions: (Koval) when integrability is only assumed near the boundary and a domain is close to an ellipse and in the situation when a domain is close to a centrally symmetric domain (Kaloshin-Koudjinan-Zhang). We also studied the question: can you deform a drum so that its sound does not deform? and what is the relation between the length spectrum and the Laplace spectrum. We made some progress for domains close to ellipses (Koval, Kaloshin-Koval-Vig). Development of analytic techniques to answer if the sound is the same or not is very important to understand the dependence of sound on the shape of the obstacle. We are confident that we can extend our knowledge or rigidity to other classes of domains as well as classes of geodesic flows on surfaces.
We established Birkhoff Conjecture in two important directions: assuming integrability only near the boundary for nearly elliptic domains and assuming that an underlying domain is close to nearly centrally symmetric. We also see a significant promise in techniques that we are developing for hyperbolic systems and for nonperturbative settings for billiards. We are confident of new significant development to come in the second part of our ERC project.
We also see a significant promise in techniques that we are developing for hyperbolic systems and for nonperturbative settings for billiards. We are confident of new significant development to come in the second part of our ERC project. For example, we have a strong belief that we can prove spectral rigidity for generic axis-symmetric billiards using combination of KAM and analysis of polygonal periodic orbits approach.
Here is a short report of each publication and preprint preparted in the framework of the
proposal so far:
1. J De Simoi, V Kaloshin, M Leguil. Marked Length Spectral determination of analytic chaotic billiards
with axial symmetries, Inventiones mathematicae 233, 829-901, 2023
2. Koudjinan E, Kaloshin V. 2022. On some invariants of Birkhoff billiards under conjugacy. Regular and
Chaotic Dynamics. 27(6), 525–537.
3. Ilya Koval, Local strong Birkhoff conjecture and local spectral rigidity of almost every ellipse, 69pp
https://arxiv.org/pdf/2111.12171 to appear in Inventiones
4. Vadim Kaloshin, Edmond Koudjinan, Ke Zhang, Birkhoff Conjecture for nearly centrally symmetric
domains https://arxiv.org/pdf/2306.12301 Geometric and Functional Analysis asked for a revision
5. Corentin Fierobe, Vadim Kaloshin, and Alfonso Sorrentino, Lecture Notes on Birkhoff Billiards: 2
Dynamics, Integrability and Spectral Rigidity, 60pp to appear in chapter in Modern Aspects of
Dynamical Systems
6. Otto Vaughn Osterman, On Length Spectrum Rigidity of Dispersing Billiard Systems Journal of Modern
Dynamics 2023, vol 19, 847-878 https://arxiv.org/pdf/2208.12244
7. Joscha Henheik, Deformational rigidity of integrable metrics on the torus, 45pp to appear in Ergodic
Theory and Dynamical Systems https://arxiv.org/abs/2210.02961
8. Vadim Kaloshin, Ilya Koval, Amir Vig, Balian-Bloch Wave Invariants for Nearly Degenerate
Orbitshttps://arxiv.org/pdf/2404.02381 28pp submitted
9. Cornetin Fiedrobe, Only quadrics have pseudo-caustics -- on caustics of Riemannian, pseudo-Euclidean
and projective billiards in higher dimensions, 22pp submitted
10. Ilya Koval, Billiard tables with analytic Birkhoff normal form are generically Gevrey divergent,
https://arxiv.org/pdf/2403.14448 65pp, submitted
11. Kostya Drach, Vadim Kaloshin, Lyapunov spectral rigidity of expanding circle maps, pre-print 28pp
In 1 we proved that the marked length spectrum determines Z2 x Z2 symmetric collection of
three analytic obstacles. A new recovery method was developed.
In 2 we analysed an old question of Guillemin asking if knowledge that two billiard maps are
conjugate we can conclude that the billiard tables are homothetic. We made partial progress
on this old question.
In 3 Koval made an important progress on Birkhoff Conjecture, where integrability is imposed
only near the boundary. The result says that under this mild integrability condition a domain
must be an ellipse.
In 4 combining techniques of Bialy-Mironov of nonstandard generating functions and
Kaloshin-Sorrentino of rational iso-integrability condition we proved that any domain close
to a centrally symmetric domain with integrable billiard is an ellipse.
In 5 we have prepared an expensive set of lecture notes about recent progress on planar
billiards.
In 6 Otto made considerable progress on recovering analytic obstacles from the marked
length spectrum.
In 7 Henheik made progress on proving that an integrable deformation of the integrable
Liouville metric is a Liouville metric.
In 8 authors made progress on showing that the singular support of the wave trace is not
equal to the length spectrum.
In 9 Fierobe studied pseudo-caustics for Riemannian, pseudo-Euclidean and projective
billiards.
In 10 Koval showed that very often if a period two billiard orbit has analytic Birkhoff Normal
form, then the table is not analytic, but only Gerver.
In 11 Drach-Kaloshin proved that for expanding circle maps satisfying sparsity condition we
have rigidity in the form that nearby expanding circle maps are smoothly conjugate.
Here is a short report of each publication and preprint preparted in the framework of the
proposal so far:
1. J De Simoi, V Kaloshin, M Leguil. Marked Length Spectral determination of analytic chaotic billiards
with axial symmetries, Inventiones mathematicae 233, 829-901, 2023
2. Koudjinan E, Kaloshin V. 2022. On some invariants of Birkhoff billiards under conjugacy. Regular and
Chaotic Dynamics. 27(6), 525–537.
3. Ilya Koval, Local strong Birkhoff conjecture and local spectral rigidity of almost every ellipse, 69pp
https://arxiv.org/pdf/2111.12171 to appear in Inventiones
4. Vadim Kaloshin, Edmond Koudjinan, Ke Zhang, Birkhoff Conjecture for nearly centrally symmetric
domains https://arxiv.org/pdf/2306.12301 Geometric and Functional Analysis asked for a revision
5. Corentin Fierobe, Vadim Kaloshin, and Alfonso Sorrentino, Lecture Notes on Birkhoff Billiards: 2
Dynamics, Integrability and Spectral Rigidity, 60pp to appear in chapter in Modern Aspects of
Dynamical Systems
6. Otto Vaughn Osterman, On Length Spectrum Rigidity of Dispersing Billiard Systems Journal of Modern
Dynamics 2023, vol 19, 847-878 https://arxiv.org/pdf/2208.12244
7. Joscha Henheik, Deformational rigidity of integrable metrics on the torus, 45pp to appear in Ergodic
Theory and Dynamical Systems https://arxiv.org/abs/2210.02961
8. Vadim Kaloshin, Ilya Koval, Amir Vig, Balian-Bloch Wave Invariants for Nearly Degenerate
Orbitshttps://arxiv.org/pdf/2404.02381 28pp submitted
9. Cornetin Fiedrobe, Only quadrics have pseudo-caustics -- on caustics of Riemannian, pseudo-Euclidean
and projective billiards in higher dimensions, 22pp submitted
10. Ilya Koval, Billiard tables with analytic Birkhoff normal form are generically Gevrey divergent,
https://arxiv.org/pdf/2403.14448 65pp, submitted
11. Kostya Drach, Vadim Kaloshin, Lyapunov spectral rigidity of expanding circle maps, pre-print 28pp
In 1 we proved that the marked length spectrum determines Z2 x Z2 symmetric collection of
three analytic obstacles. A new recovery method was developed.
In 2 we analysed an old question of Guillemin asking if knowledge that two billiard maps are
conjugate we can conclude that the billiard tables are homothetic. We made partial progress
on this old question.
In 3 Koval made an important progress on Birkhoff Conjecture, where integrability is imposed
only near the boundary. The result says that under this mild integrability condition a domain
must be an ellipse.
In 4 combining techniques of Bialy-Mironov of nonstandard generating functions and
Kaloshin-Sorrentino of rational iso-integrability condition we proved that any domain close
to a centrally symmetric domain with integrable billiard is an ellipse.
In 5 we have prepared an expensive set of lecture notes about recent progress on planar
billiards.
In 6 Otto made considerable progress on recovering analytic obstacles from the marked
length spectrum.
In 7 Henheik made progress on proving that an integrable deformation of the integrable
Liouville metric is a Liouville metric.
In 8 authors made progress on showing that the singular support of the wave trace is not
equal to the length spectrum.
In 9 Fierobe studied pseudo-caustics for Riemannian, pseudo-Euclidean and projective
billiards.
In 10 Koval showed that very often if a period two billiard orbit has analytic Birkhoff Normal
form, then the table is not analytic, but only Gerver.
In 11 Drach-Kaloshin proved that for expanding circle maps satisfying sparsity condition we
have rigidity in the form that nearby expanding circle maps are smoothly conjugate.