### Abstract

We study an area preserving map of the exterior of a smooth convex curve in the hyperbolic plane, defined by a natural geometrical construction and called the dual billiard map. We consider two problems: stability and the area spectrum. The dual billiard map is called stable if all its orbits are bounded. We show that both stable and unstable behaviours may occur. If the map at infinity has a hyperbolic periodic orbit, then the dual billiard map has orbits escaping to infinity. On the other extreme, if the map at infinity is smoothly conjugated to a Diophantine irrational rotation of the circle, then the dual billiard map is stable. The area spectrum is the set of extremal areas of n-gons, circumscribed about the dual billiard curve; this is to the dual billiard what the length spectrum is to the usual, inner, one. We show that the area spectrum has an asymptotic expansion in even negative powers of n as n → ∞. The first coefficient of this expansion is the area of the dual billiard curve, and the next is, up to a constant, the cubed integral of the cube root of its curvature. We describe the curves that are relative extrema of these two functional and show that they are the trajectories of the pseudospherical pendulum with various gravity directions.

Original language | English (US) |
---|---|

Pages (from-to) | 1051-1072 |

Number of pages | 22 |

Journal | Nonlinearity |

Volume | 15 |

Issue number | 4 |

DOIs | |

State | Published - Jul 1 2002 |

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### All Science Journal Classification (ASJC) codes

- Statistical and Nonlinear Physics
- Mathematical Physics
- Physics and Astronomy(all)
- Applied Mathematics

### Cite this

*Nonlinearity*,

*15*(4), 1051-1072. https://doi.org/10.1088/0951-7715/15/4/305

}

*Nonlinearity*, vol. 15, no. 4, pp. 1051-1072. https://doi.org/10.1088/0951-7715/15/4/305

**Dual billiards in the hyperbolic plane.** / Tabachnikov, Serge.

Research output: Contribution to journal › Article

TY - JOUR

T1 - Dual billiards in the hyperbolic plane

AU - Tabachnikov, Serge

PY - 2002/7/1

Y1 - 2002/7/1

N2 - We study an area preserving map of the exterior of a smooth convex curve in the hyperbolic plane, defined by a natural geometrical construction and called the dual billiard map. We consider two problems: stability and the area spectrum. The dual billiard map is called stable if all its orbits are bounded. We show that both stable and unstable behaviours may occur. If the map at infinity has a hyperbolic periodic orbit, then the dual billiard map has orbits escaping to infinity. On the other extreme, if the map at infinity is smoothly conjugated to a Diophantine irrational rotation of the circle, then the dual billiard map is stable. The area spectrum is the set of extremal areas of n-gons, circumscribed about the dual billiard curve; this is to the dual billiard what the length spectrum is to the usual, inner, one. We show that the area spectrum has an asymptotic expansion in even negative powers of n as n → ∞. The first coefficient of this expansion is the area of the dual billiard curve, and the next is, up to a constant, the cubed integral of the cube root of its curvature. We describe the curves that are relative extrema of these two functional and show that they are the trajectories of the pseudospherical pendulum with various gravity directions.

AB - We study an area preserving map of the exterior of a smooth convex curve in the hyperbolic plane, defined by a natural geometrical construction and called the dual billiard map. We consider two problems: stability and the area spectrum. The dual billiard map is called stable if all its orbits are bounded. We show that both stable and unstable behaviours may occur. If the map at infinity has a hyperbolic periodic orbit, then the dual billiard map has orbits escaping to infinity. On the other extreme, if the map at infinity is smoothly conjugated to a Diophantine irrational rotation of the circle, then the dual billiard map is stable. The area spectrum is the set of extremal areas of n-gons, circumscribed about the dual billiard curve; this is to the dual billiard what the length spectrum is to the usual, inner, one. We show that the area spectrum has an asymptotic expansion in even negative powers of n as n → ∞. The first coefficient of this expansion is the area of the dual billiard curve, and the next is, up to a constant, the cubed integral of the cube root of its curvature. We describe the curves that are relative extrema of these two functional and show that they are the trajectories of the pseudospherical pendulum with various gravity directions.

UR - http://www.scopus.com/inward/record.url?scp=0041929644&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=0041929644&partnerID=8YFLogxK

U2 - 10.1088/0951-7715/15/4/305

DO - 10.1088/0951-7715/15/4/305

M3 - Article

AN - SCOPUS:0041929644

VL - 15

SP - 1051

EP - 1072

JO - Nonlinearity

JF - Nonlinearity

SN - 0951-7715

IS - 4

ER -