arXiv subjects

Illya Koval

Publications and source records attributed to Illya Koval.

6 recordsLinked to original sources

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↗

arXiv2024

Exponentially many graphs are determined by their spectrum

As a discrete analogue of Kac's celebrated question on "hearing the shape of a drum", and towards a practical graph isomorphism test, it is of interest to understand which graphs are determined up to isomorphism by their spectrum (of their adjacency matrix). A striking conjecture in this area, due to van Dam and Haemers, is that "almost all graphs are determined by their spectrum", meaning that the fraction of unlabelled $n$-vertex graphs which are determined by their spectrum converges to $1$ as $n\to\infty$. In this paper we make a step towards this conjecture, showing that there are exponentially many $n$-vertex graphs which are determined by their spectrum. This improves on previous bounds (of shape $e^{c\sqrt{n}}$). We also propose a number of further directions of research.

math.CO↗

arXiv2024

Silent Orbits and Cancellations in the Wave Trace

This paper shows that the wave trace of a bounded and strictly convex planar domain may be arbitrarily smooth in a neighborhood of some point in the length spectrum. In other words, the Poisson relation, which asserts that the singular support of the wave trace is contained in the closure of $\pm$ the length spectrum, can almost be made into a strict inclusion. To do so, we construct large families of domains for which there exist multiple periodic billiard orbits having the same length but different Maslov indices. Using the microlocal Balian-Bloch-Zelditch parametrix for wave invariants developed in our previous paper, we solve a large system of equations for the boundary curvature jets, which leads to the required cancellations. We call such periodic orbits silent, since they are undetectable from the ostensibly audible wave trace. Such cancellations show that there are potential limitations in using the wave trace for inverse spectral problems and more fundamentally, that the Laplace spectrum and length spectrum are inherently different mathematical objects, at least insofar as the wave trace is concerned.

math.SP↗

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↗

arXiv2024

Billiard tables with analytic Birkhoff normal form are generically Gevrey divergent

The problem of the existence of an analytic normal form near an equilibrium point of an area-preserving map and analyticity of the associated coordinate change is a classical problem in dynamical systems going back to Poincaré and Siegel. One important class of examples of area-preserving maps consists of the collision maps for planar billiards. Recently, Treschev discovered a formal $\mathbb{Z}_2 \times \mathbb{Z}_2$ symmetric billiard with locally linearizable dynamics and conjectured its convergence. Since then, a Gevrey regularity for such a billiard was proven by Wang and Zhang, but the original problem about analyticity still remains open. We extend the class of billiards by relaxing the symmetry condition and allowing conjugacies to non-linear analytic integrable normal forms. To keep the formal solution unique, odd table derivatives and the normal form are treated as parameters of the problem. We show that for the new problem, the series of the billiard table diverge for general parameters by proving the optimality of Gevrey bounds. The general parameter set is prevalent (in a certain sense has full measure) and it contains an open set. Instead of considering the problem in a functional sense and iterating approximation procedures, we employ formal power series methods and one-by-one directly reconstruct all the Taylor coefficients of the table. In order to prove that on an open set Taylor series diverges we define a Taylor recurrence operator and prove that it has a cone property. All solutions in that cone are only Gevrey regular and not analytic.

math.DS↗

arXiv2023

Local strong Birkhoff conjecture and local spectral rigidity of almost every ellipse

The Birkhoff conjecture says that the boundary of a strictly convex integrable billiard table is necessarily an ellipse. In this article, we consider a stronger notion of integrability, namely, integrability close to the boundary, and prove a local version of this conjecture: a small perturbation of almost every ellipse that preserves integrability near the boundary, is itself an ellipse. We apply this result to study local spectral rigidity of ellipses using the connection between the wave trace of the Laplacian and the dynamics near the boundary and establish rigidity for almost all of them.

math.DS↗