### Abstract

In this paper we count closed orbits of a hyperbolic diffeomorphism restricted to a basic set. In fact we shall be dealing with two types of extensions of a hyperbolic diffeomorphism: a finite group extension (also called G-extensions) and also an automorphism extension (also called (G, ρ)-extensions). In particular we present Chebotarev type theorems and prime orbit theorem for such diffeomorphisms. These counting results are in the form of an asymptotic formula derived in complete analogy with the number theoretic result. The main difficulty is extending the domain of analyticity of the zeta and L-functions and this is overcome by resorting to symbolic dynamics. Unlike the case of a flow, the proof of this extension result does not rely on the properties of the Ruelle-Perron-Frobenius operator. Also the counting results do not use any tauberian theorems.

Original language | English |
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Pages (from-to) | 241-257 |

Number of pages | 17 |

Journal | Results in Mathematics |

Volume | 50 |

Issue number | 3-4 |

DOIs | |

Publication status | Published - Aug 2007 |

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### Keywords

- Asymptotic formulae
- Closed orbits
- Hyperbolic diffeomorphisms

### ASJC Scopus subject areas

- Mathematics (miscellaneous)
- Applied Mathematics

### Cite this

**Counting closed orbits of hyperbolic diffeomorphisms.** / Md. Noorani, Mohd. Salmi.

Research output: Contribution to journal › Article

*Results in Mathematics*, vol. 50, no. 3-4, pp. 241-257. https://doi.org/10.1007/s00025-007-0250-8

}

TY - JOUR

T1 - Counting closed orbits of hyperbolic diffeomorphisms

AU - Md. Noorani, Mohd. Salmi

PY - 2007/8

Y1 - 2007/8

N2 - In this paper we count closed orbits of a hyperbolic diffeomorphism restricted to a basic set. In fact we shall be dealing with two types of extensions of a hyperbolic diffeomorphism: a finite group extension (also called G-extensions) and also an automorphism extension (also called (G, ρ)-extensions). In particular we present Chebotarev type theorems and prime orbit theorem for such diffeomorphisms. These counting results are in the form of an asymptotic formula derived in complete analogy with the number theoretic result. The main difficulty is extending the domain of analyticity of the zeta and L-functions and this is overcome by resorting to symbolic dynamics. Unlike the case of a flow, the proof of this extension result does not rely on the properties of the Ruelle-Perron-Frobenius operator. Also the counting results do not use any tauberian theorems.

AB - In this paper we count closed orbits of a hyperbolic diffeomorphism restricted to a basic set. In fact we shall be dealing with two types of extensions of a hyperbolic diffeomorphism: a finite group extension (also called G-extensions) and also an automorphism extension (also called (G, ρ)-extensions). In particular we present Chebotarev type theorems and prime orbit theorem for such diffeomorphisms. These counting results are in the form of an asymptotic formula derived in complete analogy with the number theoretic result. The main difficulty is extending the domain of analyticity of the zeta and L-functions and this is overcome by resorting to symbolic dynamics. Unlike the case of a flow, the proof of this extension result does not rely on the properties of the Ruelle-Perron-Frobenius operator. Also the counting results do not use any tauberian theorems.

KW - Asymptotic formulae

KW - Closed orbits

KW - Hyperbolic diffeomorphisms

UR - http://www.scopus.com/inward/record.url?scp=39049140750&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=39049140750&partnerID=8YFLogxK

U2 - 10.1007/s00025-007-0250-8

DO - 10.1007/s00025-007-0250-8

M3 - Article

AN - SCOPUS:39049140750

VL - 50

SP - 241

EP - 257

JO - Results in Mathematics

JF - Results in Mathematics

SN - 1422-6383

IS - 3-4

ER -