### Abstract

We consider distribution results for closed orbits of the partially hyperbolic system: an ergodic toral automorphism Ã with respect to a (G, ρ)-extension A. In particular we obtain an analogue of the Chebotarev theorem in this situation which is an asymptotic formula for the number of closed orbits of the base transformation according to how they lift onto the extension space. To arrive at this result we introduce a cyclic extension Â of A and deduce that Â and A is essentially a group extension and homogeneous extension of Ã respectively. This observation of a group extension is similar to the setting previously studied by Parry & Pollicott and using the prime orbit theorem of Waddington we then derive at an auxiliary result for the group extension analogoues to Parry & Pollicott. Finally we relate this auxiliary result to the homogeneous extension by resorting to the work of Noorani & Parry.

Original language | English |
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Pages (from-to) | 275-285 |

Number of pages | 11 |

Journal | Bulletin of the Brazilian Mathematical Society |

Volume | 34 |

Issue number | 2 |

DOIs | |

Publication status | Published - Jul 2003 |

### Fingerprint

### Keywords

- (G, ρ)-extensions
- Chebotarev theorem
- Ergodic toral automorphisms
- Lifting closed orbits

### ASJC Scopus subject areas

- Mathematics(all)

### Cite this

**The Chebotarev theorem for ergodic toral automorphisms with respect to (G, ρ)-extensions.** / Md. Noorani, Mohd. Salmi.

Research output: Contribution to journal › Article

}

TY - JOUR

T1 - The Chebotarev theorem for ergodic toral automorphisms with respect to (G, ρ)-extensions

AU - Md. Noorani, Mohd. Salmi

PY - 2003/7

Y1 - 2003/7

N2 - We consider distribution results for closed orbits of the partially hyperbolic system: an ergodic toral automorphism Ã with respect to a (G, ρ)-extension A. In particular we obtain an analogue of the Chebotarev theorem in this situation which is an asymptotic formula for the number of closed orbits of the base transformation according to how they lift onto the extension space. To arrive at this result we introduce a cyclic extension Â of A and deduce that Â and A is essentially a group extension and homogeneous extension of Ã respectively. This observation of a group extension is similar to the setting previously studied by Parry & Pollicott and using the prime orbit theorem of Waddington we then derive at an auxiliary result for the group extension analogoues to Parry & Pollicott. Finally we relate this auxiliary result to the homogeneous extension by resorting to the work of Noorani & Parry.

AB - We consider distribution results for closed orbits of the partially hyperbolic system: an ergodic toral automorphism Ã with respect to a (G, ρ)-extension A. In particular we obtain an analogue of the Chebotarev theorem in this situation which is an asymptotic formula for the number of closed orbits of the base transformation according to how they lift onto the extension space. To arrive at this result we introduce a cyclic extension Â of A and deduce that Â and A is essentially a group extension and homogeneous extension of Ã respectively. This observation of a group extension is similar to the setting previously studied by Parry & Pollicott and using the prime orbit theorem of Waddington we then derive at an auxiliary result for the group extension analogoues to Parry & Pollicott. Finally we relate this auxiliary result to the homogeneous extension by resorting to the work of Noorani & Parry.

KW - (G, ρ)-extensions

KW - Chebotarev theorem

KW - Ergodic toral automorphisms

KW - Lifting closed orbits

UR - http://www.scopus.com/inward/record.url?scp=0141460839&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=0141460839&partnerID=8YFLogxK

U2 - 10.1007/s00574-003-0013-4

DO - 10.1007/s00574-003-0013-4

M3 - Article

AN - SCOPUS:0141460839

VL - 34

SP - 275

EP - 285

JO - Bulletin of the Brazilian Mathematical Society

JF - Bulletin of the Brazilian Mathematical Society

SN - 1678-7544

IS - 2

ER -