### Abstract

We give lower bounds on the number of periodic trajectories in strictly convex smooth billiards in R^{m+1} for m ≥ 3. For plane billiards (when m = 1) such bounds were obtained by Birkhoff in the 1920s. Our proof is based on topological methods of calculus of variations - equivariant Morse and Lusternik-Schnirelman theories. We compute the equivariant cohomology ring of the cyclic configuration space of the sphere S^{m}, i.e., the space of n-tuples of points (x_{1},...,x_{n}), where x_{i} ∈ S^{m} and x_{i} ≠ x_{i+1} for i = 1,...,n.

Original language | English (US) |
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Pages (from-to) | 553-589 |

Number of pages | 37 |

Journal | Topology |

Volume | 41 |

Issue number | 3 |

DOIs | |

State | Published - Feb 2 2002 |

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### All Science Journal Classification (ASJC) codes

- Geometry and Topology

### Cite this

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*Topology*, vol. 41, no. 3, pp. 553-589. https://doi.org/10.1016/S0040-9383(01)00021-0

**Topology of cyclic configuration spaces and periodic trajectories of multi-dimensional billiards.** / Farber, Michael; Tabachnikov, Sergei.

Research output: Contribution to journal › Article

TY - JOUR

T1 - Topology of cyclic configuration spaces and periodic trajectories of multi-dimensional billiards

AU - Farber, Michael

AU - Tabachnikov, Sergei

PY - 2002/2/2

Y1 - 2002/2/2

N2 - We give lower bounds on the number of periodic trajectories in strictly convex smooth billiards in Rm+1 for m ≥ 3. For plane billiards (when m = 1) such bounds were obtained by Birkhoff in the 1920s. Our proof is based on topological methods of calculus of variations - equivariant Morse and Lusternik-Schnirelman theories. We compute the equivariant cohomology ring of the cyclic configuration space of the sphere Sm, i.e., the space of n-tuples of points (x1,...,xn), where xi ∈ Sm and xi ≠ xi+1 for i = 1,...,n.

AB - We give lower bounds on the number of periodic trajectories in strictly convex smooth billiards in Rm+1 for m ≥ 3. For plane billiards (when m = 1) such bounds were obtained by Birkhoff in the 1920s. Our proof is based on topological methods of calculus of variations - equivariant Morse and Lusternik-Schnirelman theories. We compute the equivariant cohomology ring of the cyclic configuration space of the sphere Sm, i.e., the space of n-tuples of points (x1,...,xn), where xi ∈ Sm and xi ≠ xi+1 for i = 1,...,n.

UR - http://www.scopus.com/inward/record.url?scp=0036153754&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=0036153754&partnerID=8YFLogxK

U2 - 10.1016/S0040-9383(01)00021-0

DO - 10.1016/S0040-9383(01)00021-0

M3 - Article

VL - 41

SP - 553

EP - 589

JO - Topology

T2 - Topology

JF - Topology

SN - 0040-9383

IS - 3

ER -