
“Diamond structures in KAM invariant curves of analytical billiard-like maps”
Martedì 4 Febbraio 2025, ore 11:30 - Aula 2AB45 - Corentin Fierobe (Roma Tor Vergata)
Abstract
Mathematical billiards in strictly convex domains with smooth boundaries serve as concrete examples of twist maps on the cylinder, where the dynamics exhibit “almost integrable” behavior near the boundary of the domain. Building on this framework, Lazutkin established the existence of a Cantor set of positive measure that includes zero, within which the billiard maps feature invariant curves corresponding to certain rotation numbers. Furthermore, these invariant curves evolve smoothly as the rotation number changes, in a Whitney sense. In this presentation, I will discuss a generalization of this result for billiards with analytic boundaries, a joint work with Frank Trujillo and Vadim Kaloshin, inspired by recent contributions from Carminati, Marmi, Sauzin, and Sorrentino. This extension shows that the Cantor set of rotation numbers can be extended into the complex plane, with its complex counterpart containing structures known as “diamonds”. This finding opens up new perspectives on length spectral rigidity.