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Michael Bialy

Publications and source records attributed to Michael Bialy.

7 recordsLinked to original sources
arXiv2023

Integrable Outer billiards and rigidity

In the present paper we introduce a new generating function for outer billiards in the plane. Using this generating function, we prove the following rigidity result: if the vicinity of the smooth convex plane curve $γ$ of positive curvature is foliated by continuous curves which are invariant under outer billiard map, then the curve $γ$ must be an ellipse. In addition to the new generating function used in the proof, we also overcome the noncompactness of the phase space by finding suitable weights in the integral-geometric part of the proof. Thus, we reduce the result to the Blaschke-Santalo inequality.

math.DS↗
arXiv2022

Mather $β$-function for ellipses and rigidity

The goal of the first part of this note is to get an explicit formula for rotation number and Mather $β$-function for ellipse. This is done here with the help of non-standard generating function of billiard problem. In this way the derivation especially simple. In the second part we discuss application of Mather $β$-function to rigidity problem.

math.DS↗
arXiv2018

Billiard characterization of spheres

In this note we study the higher dimensional convex billiards satisfying the so-called Gutkin property. A convex hypersurface $S$ satisfies this property if any chord $[p,q]$ which forms angle $δ$ with the tangent hyperplane at $p$ has the same angle $δ$ with the tangent hyperplane at $q$. Our main result is that the only convex hypersurface with this property in $\mathbf{R}^d, d\geq 3$ is a round sphere. This extends previous results on Gutkin billiards obtained in \cite{B0}.

math.DG↗
arXiv2017

Gutkin billiard tables in higher dimensions and rigidity

E. Gutkin found a remarkable class of convex billiard tables in the plane which have a constant angle invariant curve. In this paper we prove that in dimension 3 only round sphere has such a property. For dimension greater than 3 it must be either a sphere or to have a very special geometric properties. In 2-dimensional case we prove a rigidity result for Gutkin billiard tables. This is done with the help of a new generating function introduced recently for billiards in our joint paper with A.E. Mironov. A formula for this generating function in higher dimensions is found.

math.DG↗
arXiv2016

Angular Billiard and Algebraic Birkhoff conjecture

In this paper we introduce a new dynamical system which we call Angular billiard. It acts on the exterior points of a convex curve in Euclidean plane. In a neighborhood of the boundary curve this system turns out to be dual to the Birkhoff billiard. Using this system we get new results on algebraic Birkhoff conjecture on integrable billiards.

math.DG↗
arXiv2008

Maximizing orbits for higher dimensional convex billiards

The main result of this paper is, that for convex billiards in higher dimensions, in contrast with 2D case, for every point on the boundary and for every $n$ there always exist billiard trajectories developing conjugate points at the $n$-th collision with the boundary. We shall explain that this is a consequence of the following variational property of the billiard orbits in higher dimension. If a segment of an orbit is locally maximizing, then it can not pass too close to the boundary. This fact follows from the second variation formula for the Length functional. It turns out that this formula behaves differently with respect to "longitudinal" and "transversal" variations.

math.DS↗