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The Chebotarov theorem for Galois coverings of Axiom A flows. (English) Zbl 0626.58006
The main result gives an analogue of Chebotarev’s theorem which provides an asymptotic formula for the number of closed orbits whose Frobenius class is a given conjugacy class $$C$$. Namely it is proved:
Theorem. Let $${\tilde \phi}$$, $$\phi$$ be axiom A flows with $${\tilde \phi}^ a G$$-covering of $$\phi$$, where $$G$$ is a Galois group. Let $${\tilde \Omega}$$ be a $${\tilde \phi}$$ basic set which is $$G$$-invariant, and let $$\Omega ={\tilde \Omega}/G$$. Define $\pi (t)=\text{Card}\{\tau | \quad e^{\lambda (\tau)h}\leq t\},\quad \pi_ C(t)=\text{Card}\{\tau | \quad e^{\lambda (\tau)h}\leq t,\quad \gamma ({\tilde \tau})\in C\}$ where $$h$$ is the topological entropy of $$\phi$$, $$\tau$$ being of the least orbit period $$\lambda$$ ($$\tau)$$, and $$\gamma$$ ($${\tilde \tau}$$) is the Frobenius element. Then $\pi_ C(t)\sim (| C| /| G|)\pi (t).$ If $${\tilde \phi}$$, $$\phi$$ are weak-mixing then $$\pi$$ (t)$$\sim t/\log t$$. An asymptotic formula is given if $${\tilde \phi}$$, $$\phi$$ are not weak-mixing.
An application to homology is given relatively to a question posed by J. Plante. Namely it is proved that the set of closed orbits with $$\Omega =M$$ generates the group $$H_ 1(M,{\mathbb Z}).$$
In the last section the basic method is applied to compact group extensions. So for the frame bundle flows on a Riemannian manifold $$M_ 0$$ with negative curvature one interprets the Frobenius class [$$\tau$$ ] of a closed orbit $$\tau$$ on $$M=TM_ 0$$ as the conjugacy class in $$G$$ of the holonomy associated with a closed geodesic.
Reviewer: M.Tarina

##### MSC:
 37D20 Uniformly hyperbolic systems (expanding, Anosov, Axiom A, etc.) 37C10 Dynamics induced by flows and semiflows 28D20 Entropy and other invariants 57T15 Homology and cohomology of homogeneous spaces of Lie groups
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##### References:
  Wiener, The Fourier integral and certain of its applications (1967)  DOI: 10.1112/blms/3.2.215 · Zbl 0219.58007  DOI: 10.1007/BF01451031 · Zbl 0094.05703  DOI: 10.1215/S0012-7094-76-04338-6 · Zbl 0346.10010  Gangolli, Illinois J. Math. 21 pp 1– (1977)  Brin, Ergodic theory and dynamical systems (1982)  DOI: 10.2307/1969183 · Zbl 0029.01503  Bowen, Equilibrium States and the Ergodic Theory of Anosov Diffeomorphisms (1975) · Zbl 0308.28010  DOI: 10.2307/2373793 · Zbl 0282.58009  DOI: 10.2307/2373590 · Zbl 0254.58005  DOI: 10.2307/1995565 · Zbl 0212.29201  DOI: 10.1016/0022-314X(82)90028-2 · Zbl 0499.10021  Ruelle, Thermodynamic Formalism (1978)  Rudolph, J. d’ Analyse Mathematique 34 pp 36– (1978)  Pollicott, Ergod. Th. & Dynam. Sys. 4 pp 135– (1984)  DOI: 10.2307/2038751 · Zbl 0249.58012  DOI: 10.2307/2006982 · Zbl 0537.58038  Parry, Ergod. Th. & Dynam. Sys. 4 pp 117– (1984)  DOI: 10.1007/BF02760669 · Zbl 0552.28020  Margulis, Funkt. Anal. i Ego Pril. 3 pp 89– (1969)  Selberg, J. Indian Math. Soc. 20 pp 47– (1956)
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