Convex billiards and a theorem by E. Hopf
type: Item
instance of: scholarly article
title: Convex billiards and a theorem by E. Hopf
zbMATH Open document ID: 0790.58023
DOI: 10.1007/BF02572397
author: Misha Bialy
published in: Mathematische Zeitschrift
publication date: 1994-05-03T00:00:00Z
full work available at URL: https://eudml.org/doc/174561
review text: We consider billiards in convex compact planar domains. Our main result states that if the phase space of the billiard ball map is foliated by not null-homotopic continuous invariant curves then the domain is circular. The result solves in part a Birkhoff's conjecture that only for elliptic domains billiards are integrable. We give a variational version of this result: the only billiards without conjugate points are circular billiards. The main tool of the paper is to apply the ``discrete version'' of E. Hopf's method invented for Riemannian manifolds without conjugate points.
Mathematics Subject Classification ID: 37A99
Mathematics Subject Classification ID: 58E30
Mathematics Subject Classification ID: 58E05
zbMATH DE Number: 561292
zbMATH Keywords: integrable billiards
zbMATH Keywords: billiards
zbMATH Keywords: convex compact planar domains
zbMATH Keywords: Birkhoff's conjecture
zbMATH Keywords: conjugate points
zbMATH Keywords: circular billiards
MaRDI profile type: Publication
cites work: Q5737099
cites work: Q3231775
cites work: Closed Surfaces Without Conjugate Points
cites work: Invariant manifolds, entropy and billiards; smooth maps with singularities. With the collab. of F. Ledrappier and F. Przytycki
cites work: Glancing billiards
cites work: Variational Construction of Orbits of Twist Diffeomorphisms
cites work: Converse KAM theory for symplectic twist maps
cites work: Periodic points of the billiard ball map in a convex domain
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description: scientific article; zbMATH DE number 561292