Dynamical Zeta Functions and Closed Orbits for Geodesic and Hyperbolic Flows

2007 ◽  
pp. 379-398 ◽  
Author(s):  
Mark Pollicott
Keyword(s):  
Zeta Functions ◽  
Closed Orbits ◽  
Hyperbolic Flows

scholarly journals Bowen's equidistribution theory and the Dirichlet density theorem

1984 ◽  
Vol 4 (1) ◽  
pp. 117-134 ◽  
Author(s):  
William Parry
Keyword(s):  
Asymptotic Formula ◽  
Prime Number ◽  
Zeta Functions ◽  
Prime Number Theorem ◽  
Density Theorem ◽  
Analytic Extension ◽  
Closed Orbits ◽  
Basic Set ◽  
Dirichlet Density ◽  
Extension Result

AbstractLet φ be an Axiom A flow restricted to a basic set, let g be a C∞ function and let , where λg(τ) is the g length of the closest orbit τ, λ(τ) is the period of τ and h is the topological entropy of φ. We obtain an asymptotic formula for πg which includes the ‘prime number’ theorem for closed orbits. This result generalizes Bowen's theorem on the equidistribution of closed orbits. After establishing an analytic extension result for certain zeta functions the proofs proceed by orthodox number theoretical techniques.

Error terms for closed orbits of hyperbolic flows

2001 ◽  
Vol 21 (2) ◽  
pp. 545-562 ◽  
Author(s):  
MARK POLLICOTT ◽  
RICHARD SHARP
Keyword(s):  
Error Term ◽  
Closed Orbit ◽  
Counting Function ◽  
Diophantine Condition ◽  
Weak Mixing ◽  
Closed Orbits ◽  
Hyperbolic Flows ◽  
Error Terms ◽  
Anosov Flows

In this paper we obtain a polynomial error term for the closed orbit counting function associated to certain hyperbolic flows. In the case of weak-mixing transitive Anosov flows no further conditions are required; for general weak-mixing hyperbolic flows a diophantine condition on the periods of the closed orbits is required.

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