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Olga Paris-Romaskevich

Publications and source records attributed to Olga Paris-Romaskevich.

8 recordsLinked to original sources
arXiv2026

Nuancing the unicity of $q$-rationals

We prove unicity of $q$-rational numbers up to conjugacy, using character varieties. Despite the unicity, we exhibit a two-parameter family of deformations of rationals with a modular symmetry. We prove that there are exactly two deformations which deliver the usual $q$-integers: the original $q$-rationals defined by Morier-Genoud and Ovsienko, and another new one. Although the new family can be obtained by conjugacy from the old one, new positivity properties appear. In addition, this new family provides a direct computation of the Jones polynomial of rational knots.

math.QA↗
arXiv2026

Plane geometry of $q$-rationals and Springborn Operations

We study the geometry of $q$-rational numbers, introduced by Morier-Genoud and Ovsienko, for positive real $q$. In particular, we construct and analyse the deformed Farey triangulation and the deformed modular surface. We interpret every $q$-rational geometrically as a circle, similar to the famous Ford circles. Further, we define and study new operations on $q$-rationals, the Springborn operations, which can be seen as a quadratic version of the Farey addition. Geometrically, the Springborn operations correspond to taking the homothety centers of a pair of two circles. As an application, we derive a formula for the $q$-deformed midpoint of two Farey neighbors and we consider a new $q$-deformation of Markov numbers.

math.QA↗
arXiv2026

Open problems in billiards and quantitative symplectic geometry

This document collects contributions to the Open Problem List in Billiards and Quantitative Symplectic Geometry, compiled following discussions during the workshop ``Billiards and quantitative symplectic geometry'' that took place at the University of Heidelberg on July 14--18, 2025.

math.SG↗
arXiv2021

Tiling billiards and Dynnikov's helicoid

Here are two problems. First, understand the dynamics of a tiling billiard in a cyclic quadrilateral periodic tiling. Second, describe the topology of connected components of plane sections of a centrally symmetric subsurface $S \subset \mathbb{T}^3$ of genus $3$. In this note we show that these two problems are related via a helicoidal construction proposed recently by Ivan Dynnikov. The second problem is a particular case of a classical question formulated by Sergei Novikov. The exploration of the relationship between a large class of tiling billiards (periodic locally foldable tiling billiards) and Novikov's problem in higher genus seems promising, as we show in the end of this note.

math.DS↗
arXiv2020

Automorphisms of compact K\"ahler manifolds with slow dynamics

We study the automorphisms of compact K\"ahler manifolds having slow dynamics. By adapting Gromov's classical argument, we give an upper bound on the polynomial entropy and study its possible values in dimensions $2$ and $3$. We prove that every automorphism with sublinear derivative growth is an isometry ; a counter-example is given in the $C^{\infty}$ context, answering negatively a question of Artigue, Carrasco-Olivera and Monteverde on polynomial entropy. Finally, we classify minimal automorphisms in dimension $2$ and prove they exist only on tori. We conjecture that this is true for any dimension.

math.DS↗
arXiv2019

Trees and flowers on a billiard table

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.

math.DS↗
arXiv2018

Triangle tiling billiards and the exceptional family of their escaping trajectories: circumcenters and Rauzy gasket

Consider a periodic tiling of a plane by equal triangles obtained from the equilateral tiling by a linear transformation. We study a following tiling billiard: a ball follows straight segments and bounces of the boundaries of the tiles into neighbouring tiles in such a way that the coefficient of refraction is equal to -1. We show that almost all the trajectories of such a billiard are either closed or escape linearly, and for closed trajectories we prove that their periods belong to the set 4N+2. We also give a precise description of the exceptional family of trajectories (of zero measure) : these trajectories escape non-linearly to infinity and approach fractal-like sets. We show that this exceptional family is parametrized by the famous Rauzy gasket. This proves several conjectures stated previously on triangle tiling billiards. In this work, we also give a more precise understanding of fully flipped minimal exchange transformations on 3 and 4 intervals by proving that they belong to a special hypersurface. Our proofs are based on the study of Rauzy graphs for interval exchange transformations with flips.

math.DS↗