The zeta functions of Ruelle and Selberg. I

David Fried

Annales scientifiques de l'École Normale Supérieure (1986)

  • Volume: 19, Issue: 4, page 491-517
  • ISSN: 0012-9593

How to cite

top

Fried, David. "The zeta functions of Ruelle and Selberg. I." Annales scientifiques de l'École Normale Supérieure 19.4 (1986): 491-517. <http://eudml.org/doc/82184>.

@article{Fried1986,
author = {Fried, David},
journal = {Annales scientifiques de l'École Normale Supérieure},
keywords = {Selberg zeta function; Fuchsian groups of the first kind; dynamical systems; suspension flow; Anosov automorphism; infranilmanifold},
language = {eng},
number = {4},
pages = {491-517},
publisher = {Elsevier},
title = {The zeta functions of Ruelle and Selberg. I},
url = {http://eudml.org/doc/82184},
volume = {19},
year = {1986},
}

TY - JOUR
AU - Fried, David
TI - The zeta functions of Ruelle and Selberg. I
JO - Annales scientifiques de l'École Normale Supérieure
PY - 1986
PB - Elsevier
VL - 19
IS - 4
SP - 491
EP - 517
LA - eng
KW - Selberg zeta function; Fuchsian groups of the first kind; dynamical systems; suspension flow; Anosov automorphism; infranilmanifold
UR - http://eudml.org/doc/82184
ER -

References

top
  1. [A] D. ANOSOV, Geodesic flows on closed Riemannian manifolds of negative curvature (Proc. Steklov Inst. Math., Vol. 90, 1967). Zbl0176.19101MR36 #7157
  2. [AB] M. ATIYAH and R. BOTT, Notes on the Lefschetz fixed point theorem for elliptic complexes, Harvard, 1964. Zbl0161.43101
  3. [AW] R. ADLER and B. WEISS, Similarity of automorphisms of the torus (Mem. AMS, Vol. 98, 1970). Zbl0195.06104MR41 #1966
  4. [BT] R. BOTT and L. TU, Differential forms in algebraic topology, Springer GTM 82, 1982. Zbl0496.55001MR83i:57016
  5. [B1] R. BOWEN, Symbolic dynamics for hyperbolic flows (Amer. J. Math., Vol. 95, 1973, pp. 429-460). Zbl0282.58009MR49 #4041
  6. [B2] R. BOWEN, On Axiom A diffeomorphisms (CBMS Reg. Conf. 35, AMS, Providence, 1978). Zbl0383.58010MR58 #2888
  7. [Bo] R. BOAS, Entire functions, Academic Press, 1954. Zbl0058.30201MR16,914f
  8. [E] L. EULER, Opera omnia, Teubner, 1922. 
  9. [Fr1] J. FRANKS, Anosov diffeomorphisms (Proc. Symp. Pure Math., Vol. 14, AMS, Providence 1970, pp. 133-164). Zbl0207.54304MR42 #6871
  10. [Fr2] J. FRANKS, Homology and dynamical systems (CBMS Reg. Conf., Vol. 49, AMS, Providence, 1982). Zbl0497.58018MR84f:58067
  11. [F1] D. FRIED, Homological identities for closed orbits (Inv. Math., Vol. 71, 1984, pp. 419-442). Zbl0512.58023MR84g:58088
  12. [F2] D. FRIED, Fuchsian groups and Reidemeister torsion (Contemp. Math., Vol. 53, 1986, pp. 141-163). Zbl0597.10027MR88e:58098
  13. [F3] D. FRIED, Efficiency vs. hyperbolicity on tori, (Springer LNM, Vol. 819, 1980, pp. 175-189). Zbl0455.58021MR82b:58065
  14. [F4] D. FRIED, Anosov foliations and cohomology (Erg. Th. & Dyn. Syst., Vol. 6, 1986, pp. 9-16). Zbl0602.57017MR87h:58172
  15. [F5] D. FRIED, Analytic torsion and closed geodesics on hyperbolic manifolds. (Inv. Math., Vol. 84, 1986, pp. 523-540). Zbl0621.53035MR87g:58118
  16. [F6] D. FRIED, Torsion and closed geodesics on complex hyperbolic manifolds, preprint. 
  17. [G] R. GANGOLLI, Zeta functions of Selberg's type for compact space forms of symmetric spaces of rank one (Ill. J. Math, Vol. 21, 1977, pp. 1-41). Zbl0354.33013MR58 #5524
  18. [G-F] I. GELFAND and S. FOMIN, Geodesic flows on manifolds of constant negative curvature (Usp. Mat. Nauk., Vol. 7, 1952, pp. 118-137, and AMS Translations, Vol. 1, 1955, pp. 49-65). Zbl0066.36101
  19. [Gr1] A. GROTHENDIECK, La théorie de Fredholm (Bull. Soc. Math. France, Vol. 84, 1956, pp. 319-384). Zbl0073.10101MR19,558d
  20. [Gr2] A. GROTHENDIECK, Espaces Nucléaires (Mem. AMS, Vol. 16, Providence, 1955). 
  21. [H] D. HEJHAL, The Selberg trace formula for PSL (2, R), Springer LNM, Vol. 548, 1976 and Vol. 1001, 1983. Zbl0543.10020MR55 #12641
  22. [HPS] M. HIRSCH, C. PUGH and M. SHUB, Invariant manifolds, Springer LNM, Vol. 583, 1977. Zbl0355.58009MR58 #18595
  23. [HR] G. HARDY and M. RIESZ, The general theory of Dirichlet series, Cambridge, 1952. 
  24. [Ma1] A. MANNING, Axiom A diffeomorphisms have rational zeta functions (Bull. London Math. Soc., Vol. 3, 1971, pp. 215-220). Zbl0219.58007MR44 #5982
  25. [Ma2] A. MANNING, Anosov diffeomorphisms on nilmanifolds (Proc. AMS, Vol. 38, 1973, pp. 423-426). Zbl0242.58004MR49 #8059
  26. [M] J. MILLSON, Closed geodesics and the ƞ-invariant (Annals of Math., Vol. 108, 1978, pp. 1-39). Zbl0399.58020MR58 #18620
  27. [PPo] W. PERRY and M. POLLICOTT, An analogue of the prime number theorem for closed orbits of Axiom A flows (Annals of Math., Vol. 118, 1983, pp. 573-592). Zbl0537.58038MR85i:58105
  28. [Po] M. POLLICOTT, Meromorphic extensions of generalized zeta functions (Inv. Math., Vol. 85, 1986, pp. 147-164). Zbl0604.58042MR87k:58218
  29. [Ra1] B. RANDOL, On the asymptotic distribution of closed geodesics on compact Riemann surfaces (Trans. AMS, Vol. 233, 1977, p. 241-247). MR58 #2643
  30. [Ra2] B. RANDOL, The Selberg trace formula, Chapter XI in Eigenvalues in Riemannian geometry, Academic Press, 1984. 
  31. [R] M. RATNER, Markov decomposition for an Y-flow on a three-dimensional manifold (Math. Notes, Vol. 6, pp. 880-886). Zbl0198.56804MR41 #5597
  32. [RS] D. RAY and I. SINGER, Analytic torsion for complex manifolds (Annals of Math., Vol. 98, 1973, pp. 154-177). Zbl0267.32014MR52 #4344
  33. [R1] D. RUELLE, Zeta functions for expanding maps and Anosov flows (Inv. Math., Vol. 34, 1976, pp. 231-242). Zbl0329.58014MR54 #8732
  34. [R2] D. RUELLE, Thermodynamic formalism, Addison-Wesley, Reading, 1978. Zbl0401.28016MR80g:82017
  35. [Sa] P. SARNAK, The arithmetic and geometry of some hyperbolic three manifolds (Acta Math., Vol. 151, 1983, pp. 253-295). Zbl0527.10022MR85d:11061
  36. [Sc] D. SCOTT, Selberg type zeta functions for the group of complex two by two matrices of determinant one (Math. Ann., Vol. 253, 1980, pp. 177-194). Zbl0449.22013MR82e:10050
  37. [S] A. SELBERG, Harmonic analysis and discontinuous groups in weakly symmetric Riemannian spaces with applications to Dirichlet series (J. Indian Math. Soc., Vol. 20, 1956, pp. 47-87). Zbl0072.08201MR19,531g
  38. [Si] Y. SINAI, Construction of Markov partitions (Func. Anal. Appl., Vol. 2, 1968, pp. 70-80). Zbl0194.22602
  39. [Sm] S. SMALE, Differentiable dynamical systems (Bull. AMS, Vol. 73, 1967, pp. 747-817). Zbl0202.55202MR37 #3598
  40. [T1] P. TOMTER, Anosov flows on infra-homogeneous spaces (Proc. Symp. Pure Math, T. 14, Providence, 1970, pp. 299-327). Zbl0207.54502MR43 #5552
  41. [T2] P. TOMTER, On the classification of Anosov flows (Topology, Vol. 14, 1975, pp. 179-189). Zbl0365.58013MR51 #14161
  42. [W] M. WAKAYAMA, Zeta functions of Selberg's type associated with homogeneous vector bundles (Hiroshima Math. J., Vol. 15, 1985, pp. 235-295). Zbl0592.22012MR86m:22026

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.