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

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Abstract

We consider distribution results for closed orbits of the partially hyperbolic system: an ergodic toral automorphism à 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  of A and deduce that  and A is essentially a group extension and homogeneous extension of à 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 languageEnglish
Pages (from-to)275-285
Number of pages11
JournalBulletin of the Brazilian Mathematical Society
Volume34
Issue number2
DOIs
Publication statusPublished - Jul 2003

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Automorphisms
Group Extension
Closed Orbit
Theorem
Hyperbolic Systems
Asymptotic Formula
Automorphism
Deduce
Orbit
Analogue

Keywords

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

ASJC Scopus subject areas

  • Mathematics(all)

Cite this

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abstract = "We consider distribution results for closed orbits of the partially hyperbolic system: an ergodic toral automorphism {\~A} 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 {\^A} of A and deduce that {\^A} and A is essentially a group extension and homogeneous extension of {\~A} 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.",
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