On the finite dimensionality of integrable deformations of strictly convex integrable billiard tables

TL;DR

This paper proves that smooth deformations of strictly convex integrable billiard tables that preserve integrability are constrained to a finite-dimensional space, indicating limited possible modifications.

Contribution

It establishes the finite dimensionality of the space of integrable deformations of convex billiard tables, advancing understanding of their structural stability.

Findings

Deformations preserving integrability form a finite-dimensional space.

Such deformations are tangent to a finite-dimensional manifold.

The result constrains possible shape changes maintaining integrability.

Abstract

In this paper, we show that any smooth one-parameter deformations of a strictly convex integrable billiard table $\Omega_0$ preserving the integrability near the boundary have to be tangent to a finite dimensional space passing through $\Omega_0$.