Some open problems in low dimensional dynamical systems
A Gasull - SeMA Journal, 2021 - Springer
The aim of this paper is to share with the mathematical community a list of 33 problems that I
have found along the years in my research. I believe that it is worth to think about them and …
have found along the years in my research. I believe that it is worth to think about them and …
The Birkhoff-Poritsky conjecture for centrally-symmetric billiard tables
M Bialy, AE Mironov - Annals of Mathematics, 2022 - projecteuclid.org
In this paper we prove the Birkhoff-Poritsky conjecture for centrally-symmetric C^2-smooth
convex planar billiards. We assume that the domain A between the invariant curve of 4 …
convex planar billiards. We assume that the domain A between the invariant curve of 4 …
New properties of triangular orbits in elliptic billiards
New invariants in the one-dimensional family of 3-periodic orbits in the elliptic billiard were
introduced by the authors in “Can the elliptic billiard still surprise us?”(2020) Math …
introduced by the authors in “Can the elliptic billiard still surprise us?”(2020) Math …
Fifty new invariants of N-periodics in the elliptic billiard
We introduce 50+ new invariants manifested by the dynamic geometry of N-periodics in the
Elliptic Billiard, detected with an experimental/interactive toolbox. These involve sums …
Elliptic Billiard, detected with an experimental/interactive toolbox. These involve sums …
Loci of 3-periodics in an Elliptic Billiard: Why so many ellipses?
A triangle center such as the incenter, barycenter, etc., is specified by a function thrice-and
cyclically applied on sidelengths and/or angles. Consider the 1d family of 3-periodics in the …
cyclically applied on sidelengths and/or angles. Consider the 1d family of 3-periodics in the …
One can hear the shape of ellipses of small eccentricity
H Hezari, S Zelditch - Annals of Mathematics, 2022 - projecteuclid.org
We show that if the eccentricity of an ellipse is sufficiently small, then up to isometries it is
spectrally unique among all smooth domains. We do not assume any symmetry, convexity …
spectrally unique among all smooth domains. We do not assume any symmetry, convexity …
The ballet of triangle centers on the elliptic billiard
The dynamic geometry of the family of 3-periodics in the Elliptic Billiard is mystifying.
Besides conserving perimeter and the ratio of inradius-to-circumradius, it has a stationary …
Besides conserving perimeter and the ratio of inradius-to-circumradius, it has a stationary …
[HTML][HTML] Totally integrable symplectic billiards are ellipses
L Baracco, O Bernardi - Advances in Mathematics, 2024 - Elsevier
In this paper we prove that a totally integrable strictly-convex symplectic billiard table, whose
boundary has everywhere strictly positive curvature, must be an ellipse. The proof, inspired …
boundary has everywhere strictly positive curvature, must be an ellipse. The proof, inspired …
Conformal transformations and integrable mechanical billiards
A Takeuchi, L Zhao - Advances in Mathematics, 2024 - Elsevier
In this article we explain that several integrable mechanical billiards in the plane are
connected via conformal transformations. We first remark that the free billiards in the plane …
connected via conformal transformations. We first remark that the free billiards in the plane …
On some refraction billiards
I De Blasi, S Terracini - arXiv preprint arXiv:2108.11159, 2021 - arxiv.org
The aim of this work is to continue the analysis, started in arXiv: 2105.02108, of the
dynamics of a point-mass particle $ P $ moving in a galaxy with an harmonic biaxial core, in …
dynamics of a point-mass particle $ P $ moving in a galaxy with an harmonic biaxial core, in …