λ-stability of periodic billiard orbits

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.