Closed orbits in homology classes.

*(English)*Zbl 0728.58026The authors consider a smooth, transitive and weakly mixing Anosov flow on a compact manifold. By introducing an analogue of the density theorem for prime numbers, they succeed to estimate the number of closed orbits in a homology class.

Given a surjective homomorphism \(\psi\) of \(H_ 1(X,Z)\) onto an abelian group H, for each \(\alpha\in H\) and for each positive number x, the authors set \[ \Pi(x,\alpha)= \{{\mathfrak p};\text{ closed orbits with } \psi[{\mathfrak p}]=\alpha \text{ and }\ell({\mathfrak p})<x\}, \] and \[ \pi(x,\alpha) = \text{ the cardinality of }\Pi(x,\alpha) \] where [\({\mathfrak p}]\) denotes the homology class and \(\ell ({\mathfrak p})\) the least period of \({\mathfrak p}.\)

One of the main results in the present paper is concerned with an asymptotic estimate of \(\pi\) (x,\(\alpha\)) as x goes to infinity. The authors have remarked the resemblance of this problem to a number theoretic problem, and demonstrate that an analogue of the density theorem for prime numbers holds. However, the main problem is that the “Galois group” H is possibly of infinite order, such that interesting extra phenomena appear.

Given a surjective homomorphism \(\psi\) of \(H_ 1(X,Z)\) onto an abelian group H, for each \(\alpha\in H\) and for each positive number x, the authors set \[ \Pi(x,\alpha)= \{{\mathfrak p};\text{ closed orbits with } \psi[{\mathfrak p}]=\alpha \text{ and }\ell({\mathfrak p})<x\}, \] and \[ \pi(x,\alpha) = \text{ the cardinality of }\Pi(x,\alpha) \] where [\({\mathfrak p}]\) denotes the homology class and \(\ell ({\mathfrak p})\) the least period of \({\mathfrak p}.\)

One of the main results in the present paper is concerned with an asymptotic estimate of \(\pi\) (x,\(\alpha\)) as x goes to infinity. The authors have remarked the resemblance of this problem to a number theoretic problem, and demonstrate that an analogue of the density theorem for prime numbers holds. However, the main problem is that the “Galois group” H is possibly of infinite order, such that interesting extra phenomena appear.

Reviewer: D.Savin (Montreal)

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\textit{A. Katsuda} and \textit{T. Sunada}, Publ. Math., Inst. Hautes Étud. Sci. 71, 5--32 (1990; Zbl 0728.58026)

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