No planar billiard possesses an open set of quadrilateral trajectories | Zendy

Alexey Glutsyuk | Zendy; Yury Kudryashov | Zendy

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No planar billiard possesses an open set of quadrilateral trajectories

Author(s) -

Alexey Glutsyuk,

Yury Kudryashov

Publication year - 2012

Publication title -

journal of modern dynamics

Language(s) - English

Resource type - Journals

SCImago Journal Rank - 0.668

H-Index - 25

eISSN - 1930-532X

pISSN - 1930-5311

DOI - 10.3934/jmd.2012.6.287

Subject(s) - dynamical billiards , mathematics , quadrilateral , boundary (topology) , measure (data warehouse) , conjecture , null set , domain (mathematical analysis) , piecewise , zero (linguistics) , open set , pure mathematics , mathematical analysis , lebesgue measure , set (abstract data type) , geometry , physics , linguistics , philosophy , database , finite element method , computer science , thermodynamics , programming language , lebesgue integration

The article is devoted to a particular case of Ivriĭ's conjecture on periodic orbits of billiards. The general conjecture states that the set of periodic orbits of the billiard in a domain with smooth boundary in the Euclidean space has measure zero. In this article we prove that for any domain with piecewise $C^4$-smooth boundary in the plane the set of quadrilateral trajectories of the corresponding billiard has measure zero.

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