Abstract
We present an overview of the known results in Lefschetz formulas for flows, that is, on the problem of relating the topology of a manifold to the number and nature of periodic orbits of a vector field.
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References
M. Atiyah and R. Bott, A Lefschetz point formula for elliptic complexes, Ann. of Math., 86 (1967), pp. 374–407.
M. Atiyah and R. Bott, A Lefschetz fixed point formula for elliptic complexes II, Ann. of Math., 88 (1968), pp. 451–491.
P. Baum and A. Connes, Chern character for discrete groups, A fete in Topology, Academic Press (1988).
J.Cheeger, Analytic torsion and the heat equation,Annals of Math. 109, pp. 259–322.
J.Cheeger, M. Gromov and M. Taylor, Finite propagation speed, kernel estimates for functions of the Laplace Operator and the geometry of complete Riemannian manifolds, J. Diff. Geom., 17 (1982), pp. 15–53.
P. Cohen and A. Connes, On the noncommutative torus of dimension two, Springer Proc. in Physics, 47 (1990), pp. 151–156.
A. Connes, A survey of foliations and operator algebras, Proc. Symp. Pure Math., 38 (1982), pp. 521–628.
A. Connes, Noncommutative differential geometry, Publications Mathematiques de l’IHES (1985).
R. Crew andD. Fried, Nonsingular holomorphic flows, Topology 25 No 4 (1986), pp. 471–473.
X. Dai, Adiabatic limits non multiplicativity of signature and the Leray spectral sequence,J. Amer. Math. Soc., vol 4, 2 (1991), pp. 265–321.
X. Dai, Geometric invariants and their adiabatic limits, MIT Preprint (1990).
J. Fay, Analytic torsion and Prym differentials, Ann. of Math. Studies, Princeton University Press (1981).
R. Forman, Personal communication.
D. Fried, Zeta functions of Ruelle and Selberg I, Ann. Sci. Ec. Norm. sup., 19 (1986), pp. 491–517.
D. Fried, Zeta functions of Ruelle and Selberg II,In preparation.
D. Fried, Lefschetz Formulas for flows, Contemp. Math., 58 III (1987), pp. 19–69.
D. Fried, Analytic torsion and closed geodesics on hyperbolic manifolds, Invent. Math., 84 (1986), pp. 523–540.
D. Fried, Torsion and closed geodesics on complex hyperbolic manifolds, Invent. Math., 91 (1988), pp. 31–51.
D. Fried, Periodic points of holomorphic maps, Topology 25 No 4 (1986), pp. 429–441.
P. B. Gilkey, Invariance theory, the heat equation and the Atiyah—Singer index theorem, Publish or Perish 11 (1984).
V. Guillemin and A. Uribe, Circular symmetry and the trace formula, Invent. Math., 96 (1989), pp. 385–423.
G. Hardy andE. Wright, An introduction to the theory of numbers, Oxford Univ. Press (1978).
T. Kotake, The fixed point theorem of Atiyah—Bott via parabolic operators, Comm Pure and Appl. Math., 23 (1969), pp. 789–806.
S. Laederich, -Torsion for complex manifolds and the adiabatic limit, Preprint, to be published in Comm. Math. Phys. (1992).
S. Laederich, 0—Torsion, foliations and holomorphic vector fields, Preprint, to be published in Comm Math. Phys. (1992).
P. Lochak, Personal communication.
R. B. Melrose, The Atiyah Patodi Singer index theorem, MIT Preprint (1991).
W. Muller, Analytic torsion and Reidemeister torsion of Riemannian manifolds, Adv. in Math., 28, 3 (1978), pp. 233–305.
C. Moore and C. Schochet, Global analysis on foliated spaces, MSRI Publ. 9, Springer-Verlag (1988).
D. Ray and I. Singer, R-torsion and the Laplacian on Riemannian Manifolds, Adv. in Math., 7 (1971), pp. 145–210.
D. Ray and J. SINGER, Analytic torsion for complex manifolds, Ann. of Math., 108 (1978), pp. 1–39.
D. Ruelle, Zeta functions for expanding maps and Anosov flows, Invent. Math., 34 (1976), pp. 231–242.
R. T. Seeley, Complex powers of an elliptic operator, Proc. Symp. Pure Math., 10 (1967), pp. 288–307.
R. O. Wells, Differential analysis on complex manifolds, Springer-Verlag, GTM 65 (1980).
E. Witten, Supersymmetry and Morse theory, Adv. in Math., 4 (1970), pp. 109–126.
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Laederich, S. (1995). Analytic Torsion, Flows and Foliations. In: Dumas, H.S., Meyer, K.S., Schmidt, D.S. (eds) Hamiltonian Dynamical Systems. The IMA Volumes in Mathematics and its Applications, vol 63. Springer, New York, NY. https://doi.org/10.1007/978-1-4613-8448-9_13
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DOI: https://doi.org/10.1007/978-1-4613-8448-9_13
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