Spectrum of the Poincaré map for periodic reflecting rays in generic domains
The following theorem is proved: Let X be a smooth compact (n-1)- dimensional submanifold of \(R^ n\), \(n\geq 2\), and let \({\mathcal A}\) be a countable set of complex numbers \(\mu\neq 0\). Denote by \(T_{{\mathcal A}}\) the set of those \(f\in C^{\infty}_{emb}(X,R^ n)\), \((C^{\infty}\)- embeddings of X into \(R^ n)\) such that for every periodic ray \(\gamma\) reflecting on f(X), nontangent to f(X), the spectrum of the Poincaré map \(P_{\gamma}\) related to \(\gamma\) (mapping each reflection point to the next one) has no elements in \({\mathcal A}\). Then T contains a residual subset of \(C^{\infty}_{emb}(X,R^ n)\). This is a Kupka-Smale type theorem. The proof relies on Thom's transversality theorem in jets of mappings.
scientific article; zbMATH DE number 4102141
scientific article