Closed orbits in homology classes. (English) Zbl 0728.58026

Katsuda, Atsushi; Sunada, Toshikazu

Publ. Math., Inst. Hautes Étud. Sci. 71, 5-32 (1990).

The 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.

Reviewer: D.Savin (Montreal)

MSC:

37D99	Dynamical systems with hyperbolic behavior
37A99	Ergodic theory

Keywords:

Anosov flows; closed orbits in homology classes

References:

[1]	T. Adachi, Meromorphic extension of L-functions of Anosov flows and profinite graphs,Kumamoto J. Math.,1 (1988), 9–24. · Zbl 0668.58041
[2]	T. Adachi andT. Sunada, Homology of closed geodesics in a negatively curved manifold,J. Diff. Geom.,26 (1987), 81–99. · Zbl 0618.58028
[3]	T. Adachi andT. Sunada, Twisted Perron-Frobenius theorem and L-functions,J. Funct. Anal.,71 (1987), 1–46. · Zbl 0658.58034
[4]	R. Bowen, Symbolic dynamics for hyperbolic flows,Amer. J. Math.,95 (1973), 429–459. · Zbl 0282.58009
[5]	M. Denker andW. Philipp, Approximation by Brownian motion for Gibbs measures and flows under a function,Ergod. Th. & Dynam. Syst.,4 (1984), 541–552. · Zbl 0554.60077
[6]	C. Epstein, Asymptotics for closed geodesics in a homology class–finite volume case–,Duke Math. J.,55 (1987), 717–757. · Zbl 0648.58041
[7]	I. M. Gel’fand andI. I. Pyateckii-Shapiro, A theorem of Poincaré,Dokl. Akad. Nauk.,127 (1959), 490–493 (Russian).
[8]	V. Guillemin andD. Kazhdan, Some inverse spectral results for negatively curved 2-manifolds,Topology,11 (1980), 301–312. · Zbl 0465.58027
[9]	P. H. Halmos,Measure Theory, New York, Van Nostrand Co. Inc., 1950.
[10]	A. Katsuda, Density theorem for closed orbits,Proc. of Taniguchi Symp., Springer Lect. Notes in Math., vol. 1339, 1988, to appear. · Zbl 0668.58043
[11]	A. Katsuda andT. Sunada, Homology and closed geodesics in a compact Riemann surface,Amer. J. Math.,110 (1988), 145–156. · Zbl 0647.53036
[12]	S. Lang,Algebraic Number Theory, London, Addison-Wisley, 1970. · Zbl 0211.38404
[13]	A. N. Livcic, Some homological properties of U-systems,Math. Zametki,10 (1971), 555–564.
[14]	S. Manabe andY. Ochi, The central limit theorem for current-valued processes induced by geodesic flows,Osaka J. Math.,26 (1989), 191–205. · Zbl 0703.60019
[15]	W. Parry, Bowen’s equidistribution theory and the Dirichlet density theorem,Ergod. Th. & Dynam. Syst.,4 (1984), 117–134. · Zbl 0567.58014
[16]	W. Parry andM. Pollicott, An analogue of the prime number theorem for closed orbits of Axiom A flows,Ann. of Math.,118 (1983), 573–591. · Zbl 0537.58038
[17]	W. Parry andM. Pollicott, The Chebotarev theorem for Galois coverings of Axiom A flows,Ergod. Th. & Dynam. Syst.,6 (1986), 133–148. · Zbl 0626.58006
[18]	R. Phillips andP. Sarnak, Geodesics in homology classes,Duke Math. J.,55 (1987), 287–297. · Zbl 0642.53050
[19]	J. Plante, Anosov flows,Amer. J. Math.,94 (1972), 729–758. · Zbl 0257.58007
[20]	J. Plante, Homology of closed orbits of Axiom A flows,Proc. A. M. S.,37 (1973), 297–300. · Zbl 0249.58012
[21]	M. Pollicott, On the rate of mixing of Axiom A flows,Invent. Math.,81 (1985), 413–426. · Zbl 0591.58025
[22]	D. Ruelle,Thermodynamic Formalism, London, Addison-Wisley, 1978.
[23]	A. Selberg, Harmonic analysis and discontinuous subgroups in weakly symmetric Riemannian spaces with applications to Dirichlet series,J. Indian Math. Soc.,20 (1956), 47–87. · Zbl 0072.08201
[24]	J.-P. Serre, Divisibilité de certaines fonctions arithmétiques,L’Ens. Math.,22 (1976), 227–260.
[25]	S. Zelditch,Geodesics in homology classes and periods of automorphic forms, preprint, Johns Hopkins Univ., 1986.
[26]	M. H. Delange, Généralisation du théorème de Ikehara,Ann. Scient. Ec. Norm. Sup., Sér. 3,71 (1954), 213–242. · Zbl 0056.33101