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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

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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