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Corentin Fierobe

Publications and source records attributed to Corentin Fierobe.

14 recordsLinked to original sources
arXiv2026

Rigidity of Mather's $\beta$-function on a KAM set for analytic billiards-like maps and unique quasi-analytic continuation

In his seminal paper, Kac famously asked whether "one can hear the shape of a drum" - that is whether the isometry class of a bounded domain in a Euclidean space is uniquely determined by the spectrum of its Laplace spectrum. The Laplace spectrum is closely related to the length spectrum of the associated billiard. For a convex bounded planar domain, to each billiard periodic orbit one can associate not only its length but also its rotation number. The set of pairs of length and rotation number of each periodic orbit is called the marked length spectrum. Using the marked length spectrum of the domain one can associate with it its minimal action function also known as Mathers $\beta$-function denoted by $\beta_\Omega$. Via unique quasianalytic continuation, we prove the following rigidity problem: knowing that Mather's $\beta$-functions of two billiards in two analytic planar domains coincide on a set of positive measure of KAM diophantine numbers implies that they coincide on a set of all KAM diophantine numbers. In particular, knowing that Mather's $\beta$-functions of two domains coincide on a positive measure set of KAM diophantine numbers implies that the Marvizi-Melrose invariants of these domains coincide.

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arXiv2026

Spectral rigidity among ellipses, Bialy's conjecture and local extrema of Mather's beta function

In this paper we prove Bialy's conjecture which states that if the Mather beta functions of two ellipses coincide at two nonzero rotation numbers then the ellipses coincide. We also show that the same conclusion holds when only one rotation number is prescribed, provided the two ellipses have the same perimeter. Finally we discuss consequences for local extremizers of Mathers beta function building on a recent result of Baranzini, Bialy and Sorrentino.

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arXiv2026

Rigidity of the Suris' potential in the Frenkel-Kontorova Model

The goal of this paper is to establish a local rigidity result for the integrability of standard-like maps. The main focus of the paper is the remarkable integrable potential discovered by Suris in the 80's. We show that locally, the integrability of this potential is rigid. The proof relies on a similar strategy that was used for billiards in an ellipse, and involves developing the action-angle coordinates for this system, and exploiting it to construct a Riesz basis for $L^2$. As a corollary, we obtain a spectral rigidity result for this setting. Finally, we study the integrability question in the setting of potentials that are periodic.

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arXiv2025

A billiard table close to an ellipse is deformationally spectrally rigid among dihedrally symmetric domains

We prove that a a strongly convex planar domain (Birkhoff table) with dihedral symmetry, which is sufficiently close in a finitely smooth topology to an ellipse, is deformationally spectrally rigid within the class of domains preserving this symmetry. More precisely, any smooth one-parameter family of such domains that preserves the length spectrum (i.e., the set of lengths of periodic billiard orbits) must consist only of rigid motions of the initial domain. The proof combines two types of dynamical data: the asymptotic behavior of certain symmetric periodic orbits, as previously used in the rigidity of nearly circular domains, and new spectral information derived from KAM invariant curves, obtained from Mather's beta function and its derivatives (in the Whitney sense) at some suitable rotation numbers.

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arXiv2025

Starting the study of outer length billiards

We focus on the outer length billiard dynamics, acting on the exterior of a strictly-convex planar domain. We first show that ellipses are totally integrable. We then provide an explicit representation of first order terms for the formal Taylor expansion of the corresponding Mather's $\beta$-function. Finally, we provide explicit Lazutkin coordinates up to order 4.

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arXiv2025

Lazutkin coordinates and the conjugation problem for billiard maps

The conjugation problem for billiard maps conjectures that if two strictly convex billiards have conjugated billiard maps, the billiard tables must be homothetic to each other. We show that if two billiard maps are conjugated, the conjugation diffeomorphism is tangent to a Lazutkin change of coordinates. We also recompute the coefficients in the billiard map expansion as the angle ttends to zero, correcting some incorrect expressions in Lazutkin's computations.

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arXiv2024

Deformational spectral rigidity of axially-symmetric symplectic billiards

Symplectic billiards were introduced by Albers and Tabachnikov as billiards in strictly convex bounded domains of the plane with smooth boundary having a specific law of reflection. This paper proves a rigidity result for symplectic billiards which is similar to a previous result on classical billiards formulated by De Simoi, Kaloshin and Wei. Namely, it states that close to an ellipse, a sufficiently smooth one-parameter family of axially symmetric domains either contains domains with different area-spectra or is trivial, in a sense that the domains differ by area-preserving affine transformations of the plane. The paper also prove that in the general setting - that is even if the domains are not close to an ellipse - any sufficiently smooth one-parameter family of axially symmetric domains which preserves the area-spectrum is tangent to a finite dimensionnal space.

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arXiv2024

On the existence of periodic invariant curves for analytic families of twist maps and billiards

In this paper we prove that in any analytic one-parameter family of twist maps of the annulus, homotopically invariant curves filled with periodic points corresponding to a given rotation number, either exist for all values of the parameters or at most for a discrete subset. Moreover, we show that the set of analytic twist maps having such an invariant curve of a given rotation number is a strict analytic subset of the set of analytic twist maps. The first result extends, in dimension 2, a previous result by Arnaud, Massetti and Sorrentino. We then apply our result to rational caustics of billiards, considering several models such as Birkhoff billiards, outer billiards and symplectic billiards.

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arXiv2023

Only quadrics have pseudo-caustics -- on caustics of Riemannian, pseudo-Euclidean and projective billiards in higher dimensions

This paper studies billiard models with a generalized law of reflection, the so-called projective billiards. They unify various laws, including the classical one in a Euclidean, pseudo-Euclidean or Riemannian metric. They were introduced and studied by Sergey Tabachnikov. The paper studies these models in dimension at least 3 and focuses on the existece and properties of caustics, i.e. hypersurfaces to which any light trajectory remains tangent to after successive reflections. It restricts then the study to the particular case of Riemannian billiards whose geodesics are supported by lines, and of pseudo-Euclidean billiards. More precisely, this article 1) describes the class of projective billiards whose only possible caustics are quadrics; 2) translates this result to Riemannian billiards; 3) applies it to show that if a pseudo-Euclidean billiard has a caustic, then both are pseudo-confocal quadrics.

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arXiv2020

On projective billiards with open subsets of triangular orbits

Ivrii's Conjecture states that in every billiard in Euclidean space the set of periodic orbits has measure zero. It implies that for every $k\geq2$ there are no k-reflective billiards, i.e., billiards having an open set of k-periodic orbits. This conjecture is open in Euclidean spaces, with just few partial results. It is known that in the two-dimensional sphere there exist 3-reflective billiards (Yu.M.Baryshnikov). All the 3-reflective spherical billiards were classified in a paper by V.Blumen, K.Kim, J.Nance, V.Zharnitsky: the boundary of each of them lies in three orthogonal big circles. In the present paper we study the analogue of Ivrii's Conjecture for projective billiards introduced by S.Tabachnikov. In two dimensions there exists a 3-reflective projective billiard, the so-called right-spherical billiard, which is the projection of a spherical 3-reflective billiard. We show that the only 3-reflective planar projective billiard with piecewise smooth boundary is the above-mentioned right-spherical billiard. In higher dimensions, we prove the non-existence of 3-reflective projective billiards with piecewise smooth boundary, and also the non-existence of projective billiards with piecewise smooth boundary having a subset of triangular orbits of non-zero measure in the phase space.

math.DS↗
arXiv2020

Examples of reflective projective billiards and outer ghost billiards

In the class of projective billiards, which contains the usual billiards, we exhibit counter-examples to Ivrii's conjecture, which states that in any planar billiard with smooth boundary the set of periodic orbits has zero measure. The counter-examples are polygons admitting a $2$-parameters family of $n$-periodic orbits, with $n$ being either $3$ or any even number grower than $4$.

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arXiv2019

Complex caustics of the elliptic billiard

The article studies a generalization of the elliptic billiard to the complex domain. We show that the billiard orbits also have caustics, and that the number of such caustics is bigger than for the real case. For example, for a given ellipse E, there exist exactly two confocal ellipses such that the triangular orbits of E are circumscribed about one of them, and each tangent line to one of those ellipses is a side of a triangular orbit. We also give a bound on the number of caustics for orbits with a fixed number of sides.

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