$\lambda$-stability of periodic billiard orbits

TL;DR

This paper introduces a new concept of stability for periodic billiard orbits called $mbda$-stability, providing conditions for its occurrence and characterizing such orbits in integrable polygons.

Contribution

It defines $mbda$-stability for periodic billiard orbits, establishes criteria for stability, and characterizes stable orbits in integrable polygons, expanding understanding of billiard dynamics.

Findings

Set of polygons with finitely many $mbda$-stable orbits is dense.

Provided necessary and sufficient conditions for $mbda$-stability.

Complete characterization of $mbda$-stable orbits in integrable polygons.

Abstract

We introduce a new notion of stability for periodic orbits in polygonal billiards. We say that a periodic orbit of a polygonal billiard is $\lambda$-stable if there is a periodic orbit for the corresponding pinball billiard which converges to it as $\lambda$ $\rightarrow$ 1. This notion of stability is unrelated to the notion introduced by Galperin, Stepin and Vorobets. We give sufficient and necessary conditions for a periodic orbit to be $\lambda$-stable and prove that the set of d-gons having at most finite number of $\lambda$-stable periodic orbits is dense is the space of d-gons. Moreover, we also determine completely the $\lambda$-stable periodic orbits in integrable polygons.