Counting closed orbits of hyperbolic diffeomorphisms

Research output: Contribution to journalArticle

4 Citations (Scopus)

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 languageEnglish
Pages (from-to)241-257
Number of pages17
JournalResults in Mathematics
Volume50
Issue number3-4
DOIs
Publication statusPublished - Aug 2007

Fingerprint

Closed Orbit
Diffeomorphisms
Counting
Orbits
Diffeomorphism
Ruelle Operator
Perron-Frobenius Operator
Tauberian theorem
Group Extension
Symbolic Dynamics
Analyticity
L-function
Theorem
Asymptotic Formula
Riemann zeta function
Automorphism
Analogy
Count
Finite Group
Orbit

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.

In: Results in Mathematics, Vol. 50, No. 3-4, 08.2007, p. 241-257.

Research output: Contribution to journalArticle

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