Gevrey regularity for the formally linearizable billiard of Treschev

TL;DR

This paper proves that Treschev's formally linearizable billiard example exhibits Gevrey regularity under Diophantine conditions, using an iterative scheme to analyze its structure and symmetries.

Contribution

It establishes Gevrey regularity for Treschev's billiard example under Diophantine frequencies, advancing understanding of its analytic properties.

Findings

The billiard example is $(1+ eta)$-Gevrey for some $eta > 0$ under Diophantine conditions.

The iterative scheme clarifies the structure and symmetries of Treschev's construction.

The result suggests potential convergence of the formal series.

Abstract

Treschev made the remarkable discovery that there exists formal power series describing a billiard with locally linearizable dynamics. We show that if the frequency for the linear dynamics is Diophanine, the Treschev example is $(1+ \alpha)$-Gevrey for some $\alpha > 0$. Our proof is based on an iterative scheme that further clarifies the structure and symmetries underlying the original Treschev construction. Hopefully, Our result sheds a light on the more important question of whether this example is convergent.