How smooth can convex chaotic billiard tables be?

TL;DR

This paper investigates the minimal boundary smoothness needed to eliminate chaos in stadium billiards, extending KAM theory to low-smoothness systems and showing that $C^2$ smoothing introduces stable periodic orbits.

Contribution

It provides the first analysis of smoothing effects on chaotic billiards with low boundary regularity, extending KAM theory to $C^1$ boundaries.

Findings

$C^2$ smoothing creates elliptic periodic orbits

The stadium's boundary smoothness critically affects chaotic dynamics

Extended KAM theory applies to low-smoothness billiard systems

Abstract

We solve the longstanding problem of smoothing a stadium billiard. Besides our results demonstrate why there were no clear conjectures how much the stadium's boundary must be smoothened to destroy chaotic dynamics. To do that we needed to extend standard KAM theory to analyze stability of periodic orbits, because of the low smoothness of the system. In fact, the stadium has a $C^1$ boundary, and we show that $C^2$ smoothing results in appearance of elliptic periodic orbits.