Searcharxiv⌕ Search

arXiv subjects

Vadim Kaloshin

Publications and source records attributed to Vadim Kaloshin.

At least 19 recordsLinked to original sources
arXiv2026

Unmarked spectral rigidity of expanding circle maps

For a smooth expanding map $f$ of the circle, its (unmarked) length spectrum is defined as the set of logarithms of multipliers of periodic orbits of $f$. This spectrum is analogous to the set of lengths of all closed geodesics on negatively curved surfaces -- the classical length spectrum. In this paper, we prove a length spectral rigidity result for expanding circle maps. Namely, in finite regularity, we show that a prevalent, and hence dense, set of expanding circle maps $f$ of degree $d \geqslant 2$ is locally rigid, that is, every nearby isospectral map is smoothly conjugate to $f$. The proof introduces a Livsic--Whitney approximation scheme for hyperbolic systems. The approximation step consists of constructing an approximate coordinate change of low regularity using a finite Livsic theorem and modifying this change using the Whitney extension theorem. This approximation scheme is reminiscent of a KAM scheme.

math.DS↗
arXiv2026

Cyclicity of lips and center-focus problem at infinity

In this work, we discover an intriguing Liénard system which is integrable near infinity, integrable near zero and having Ilyashenko-Kotova lips in between. Moreover, we give a necessary and sufficient condition of polynomial Liénard system with center at infinity. In a different direction we improve a lower bound on Hilbert number for Liénard systems. The improvement is due to development of Brudnyi method of calculation of cyclicity of centers near infinity and near zero.

math.DS↗
arXiv2026

Prevalence of sparsity for negatively curved metrics

Let $M$ be a closed manifold of arbitrary dimension. The length spectrum of a negatively curved metric $g$ on $M$ is the set of lengths of all possible closed geodesics of $g$. We prove that metrics whose length spectrum is exponentially sparse, i.e., the gaps between distinct lengths are bounded below by a quantity decaying exponentially in the length, are prevalent, and hence dense, among $C^k$-smooth negatively curved metrics on $M$ for sufficiently large $k$. Moreover, unlike in all known constructions, the exponent in our bound grows only sublinearly in the regularity $k$. This work is motivated by our recent result on prevalent unmarked length spectral rigidity for expanding circle maps and a very recent result by DeWitt, Durham, Reber, and O'Hare that $C^k$-smooth negatively curved metrics with exponentially sparse spectra are locally rigid.

math.DG↗
arXiv2026

Rigidity of Mather's $β$-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 $β$-function denoted by $β_Ω$. Via unique quasianalytic continuation, we prove the following rigidity problem: knowing that Mather's $β$-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 $β$-functions of two domains coincide on a positive measure set of KAM diophantine numbers implies that the Marvizi-Melrose invariants of these domains coincide.

math.DS↗
arXiv2026

Spectral rigidity of two-dimensional Liouville tori

It is a folklore conjecture that Liouville metrics are the only Riemannian metrics with integrable geodesic flow on the two-dimensional torus. In this paper, we study Laplace-isospectral deformations, within a fixed conformal class, of generic Liouville metrics and prove two rigidity results: First, any isospectral deformation that is affine in $ε$ within the fixed conformal class is trivial. Second, any isospectral deformation that is analytic in the perturbation parameter and whose Taylor coefficients are trigonometric polynomials in the spatial variables remains Liouville and is obtained by componentwise rearrangement of the original metric. For the first result, the proof combines a wave-trace noncancellation result, which allows us to recover suitable length data from the Laplace spectrum, with a second-variation analysis of the energy functional along closed geodesics. For the second result, we prove that, within the Liouville class considered here, the relevant length-isospectrality condition is equivalent to componentwise rearrangement.

math.DG↗
arXiv2026

Isoenergetic degeneracy generically creates meandering invariant tori

Consider the set of isoenergetically degenerate integrable Hamiltonians with two degrees of freedom. We show that a cusp-generic perturbation of a generic Hamiltonian in this set gives rise to meandering invariant tori - embedded Lagrangian tori which are not graphs. Moreover, an exponentially dense subset of perturbations admits higher order meandering tori, of all orders from two to infinity. These infinite order meanders have an endless nested structure.

math.DS↗
arXiv2026

Computation of a separatrix map and a normally hyperbolic invariant lamination for the RP3BP

In this paper we discuss the existence of a normally hyperbolic invariant lamination (NHIL) at the Kirkwood gap $3:1$ for the Restricted Planar Elliptic 3 Body Problem. This problem models the Sun-Jupiter-Asteroid dynamics. We also show that the induced dynamics on the NHIL is a partially hyperbolic skew-shift which is of the form \[ f:(ω,I,θ)\to (σω, I+e_0 A_ω(I)\cos(θ+ψ_ω)+\mathcal{O}(e^2_0), θ+Ω_ω(I)+\mathcal{O}(e_0)),\] where $I\in [a,b], θ\in \mathbb T, ω\inΣ=\{0,1\}^\mathbb Z$, the space of sequences of $0,1$'s, $σ:Σ\to Σ$ is the shift in this space, $Ω_ω$ is the shear, $A_ω$ is an amplitude, and $e_0$ is the eccentricity of Jupiter, which is taken as a small parameter. In the companion paper arXiv:2603.19894, relying on these skew-shift, we show the existence of stochastic diffusing behavior for Asteroids belonging to the Kirkwood gap provided the eccentricity of Jupiter is $e_0$ small enough. Key ingredients to construct the NHIL are the separatrix map associated to homoclinic channels to a normally hyperbolic invariant cylinder and an isolating block construction. Some of the necessary non-degeneracy conditions are verified numerically.

math.DS↗
arXiv2026

Stochastic behavior along mean motion resonances in the restricted planar 3-body problem

One of the most remarkable instability zones in the Solar system are Kirkwood gaps in the asteroid belt. In this paper we analyze instabilities in the famous Kirkwood gap $3:1$ in the regime of small eccentricity of Jupiter. Mathematically speaking, we study the evolution of asteroids under the influence of the Sun and Jupiter using the restricted planar elliptic 3-body problem (RPE3BP) for initial conditions near a mean motion resonance 3:1. The main result exhibits stochastic diffusing behavior of the eccentricity of the asteroid for a rich set of initial conditions. Roughly speaking, for small eccentricity $e_0$ of Jupiter, the evolution of the eccentricity of the asteroid $\mathbf{e}(t\cdot e_0^{-2})$ at the Kirkwood gap $3:1$ behaves like a diffusion process on the line, where the randomness comes from the initial conditions. Along with KAM theory, we have mixed behavior in the asteroid belt, that is coexistence of quasiperiodic (deterministic) and stochastic (diffusive) behavior. See also the companion paper arXiv:2603.19893

math.DS↗
arXiv2026

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.

math.DS↗
arXiv2025

Kuznecov remainders and generic metrics

We obtain improved remainder estimates in the Kuznecov sum formula for period integrals of Laplace eigenfunctions on a Riemannian manifold $M$. Building upon the two-term asymptotic expansion established in [arXiv:2204.13525], we prove that for a Baire-generic class of metrics, the oscillatory second term in the Kuznecov formula can be eliminated, yielding an improved remainder estimate.

math.AP↗
arXiv2024

Birkhoff Conjecture for nearly centrally symmetric domains

In this paper we prove a perturbative version of a remarkable Bialy-Mironov (Ann.Math:389-413(196), 2022) result. They prove non perturbative Birkhoff conjecture for centrally-symmetric convex domains, namely, a centrally-symmetric convex domain with integrable billiard is ellipse. We combine techniques from Bialy-Mironov (Ann.Math:389-413(196), 2022) with a local result by Kaloshin-Sorrentino (Ann.Math:315-380(188), 2018) and show that a domain close enough to a centrally-symmetric one with integrable billiard is ellipse. To combine these results we derive a slight extension of Bialy-Mironov (Ann.Math:389-413(196), 2022) by proving that a notion of rational integrability is equivalent to the $C^0$-integrability condition used in their paper.

math.DS↗
arXiv2024

Balian-Bloch Wave Invariants for Nearly Degenerate Orbits

This paper is part I of a series in which we aim to show that the singular support of the wave trace and the length spectrum of a smooth, strictly convex, and bounded planar billiard table are generally distinct objects. We derive an asymptotic trace formula for the regularized resolvent which is dual to the wave trace and contains the same information. To do this, we consider a class of periodic orbits which have nearly degenerate Poincaré maps and study their leading order behavior as the deformation parameter goes to zero, generating large coefficients in the wave trace. We also keep careful track of the Maslov indices, which will allow us to match contributions of opposite signs in our subsequent paper. Each cancellation of coefficients in the resolvent trace corresponds to making the wave trace one degree smoother. The resolvent based approach is due to Balian and Bloch and was significantly expanded upon by Zelditch in a foundational series of papers [Zel09], [Zel04a], [Zel04c] and [Zel00].

math.SP↗
arXiv2022

Marked Length Spectral determination of analytic chaotic billiards with axial symmetries

We consider billiards obtained by removing from the plane finitely many strictly convex analytic obstacles satisfying the non-eclipse condition. The restriction of the dynamics to the set of non-escaping orbits is conjugated to a subshift, which provides a natural labeling of periodic orbits. We show that under suitable symmetry and genericity assumptions, the Marked Length Spectrum determines the geometry of the billiard table.

math.DS↗
arXiv2022

Length Spectrum Rigidity for piecewise analytic Bunimovich Billiards

In the paper, we establish Squash Rigidity Theorem - the dynamical spectral rigidity for piecewise analytic Bunimovich squash-type stadia. We also establish Stadium Rigidity Theorem - the dynamical spectral rigidity for piecewise analytic Bunimovich stadia whose flat boundaries are a priori fixed. In addition, for smooth Bunimovich squash-type stadia we compute the Lyapunov exponents along the maximal period two orbit, as well as the value of the Peierls' Barrier function from the maximal marked length spectrum associated to the rotation number $\frac{2n}{4n+1}$.

math.DS↗
arXiv2019

Marked Length Spectrum, homoclinic orbits and the geometry of open dispersing billiards

We consider billiards obtained by removing three strictly convex obstacles satisfying the non-eclipse condition on the plane. The restriction of the dynamics to the set of non-escaping orbits is conjugated to a subshift on three symbols that provides a natural labeling of all periodic orbits. We study the following inverse problem: does the Marked Length Spectrum (i.e., the set of lengths of periodic orbits together with their labeling), determine the geometry of the billiard table? We show that from the Marked Length Spectrum it is possible to recover the curvature at periodic points of period two, as well as the Lyapunov exponent of each periodic orbit.

math.DS↗
arXiv2018

Density of convex billiards with rational caustics

We show that in the space of all convex billiard boundaries, the set of boundaries with rational caustics is dense. More precisely, the set of billiard boundaries with caustics of rotation number $1/q$ is polynomial sense in the smooth case, and exponentially dense in the analytic case.

math.DS↗