Spectrum of the Poincaré map for periodic reflecting rays in generic domains
type: Item
instance of: scholarly article
title: Spectrum of the Poincaré map for periodic reflecting rays in generic domains
zbMATH Open document ID: 0673.58035
DOI: 10.1007/BF01161919
author: Vesselin Petkov
author: Luchezar Stoyanov
published in: Mathematische Zeitschrift
publication date: 1987-01-01T00:00:00Z
full work available at URL: https://eudml.org/doc/173852
review text: 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.
Mathematics Subject Classification ID: 37G99
Mathematics Subject Classification ID: 58J50
Mathematics Subject Classification ID: 37A30
zbMATH DE Number: 4102141
zbMATH Keywords: billiards
zbMATH Keywords: generic properties
zbMATH Keywords: scattering
zbMATH Keywords: inverse spectral
zbMATH Keywords: problem
zbMATH Keywords: periodic ray
zbMATH Keywords: Kupka-Smale type theorem
zbMATH Keywords: jets
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 4102141