By Burghelea D., Haller S.

During this paper, we give some thought to vector fields on a closed manifold whose instantons and closed trajectories should be 'counted'. Vector fields which admit Lyapunov closed one types belong to this category. We express that below an extra speculation, 'the exponential progress property', the counting services of instantons and closed trajectories have Laplace transforms which might be on the topic of the topology and the geometry of the underlying manifold. the aim of this paper is to introduce and discover the concept that 'exponential progress property', and to explain those Laplace transforms.

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We conclude that T IntX η˜ 2 = eL (h ∗ P X )( η˜ ) 2 ˚ X \ ΣX ) ∩ P ˚ X . 27, holds for an open set of η˜ ∈ (R X X ˚ ˚ X . 5), it remains this relation holds for all η˜ ∈ (R \ Σ ) ∩ P X X X true for all η ∈ (R \ Σ ) ∩ P . 149 DYNAMICS, LAPLACE TRANSFORM AND SPECTRAL GEOMETRY Appendix A. 5. We will make use of the following lemma whose proof we leave to the reader. 1. Let N be a compact smooth manifold, possibly with boundary, and let K ⊆ N be a compact subset. Let L := N × ∂I ∪ K × I where I := [0, 1].

26. H. Whitney, Complex analytic varieties (Addison-Wesley, Reading, MA, 1972). 27. D. V. Widder, The Laplace transform, Princeton Mathematical Series 6 (Princeton University Press, Princeton, NJ, 1941).

Austin and P. J. Braam, ‘Morse–Bott theory and equivariant cohomology’, The Floer memorial volume, Progress in Mathematics 133 (Birkh¨a user, Basel, 1995) 123–183. 2. J. M. Bismut and W. Zhang, ‘An extension of a theorem by Cheeger and M¨ u ller’, Ast´erisque 205 (Soci´et´e Math´ematique de France, Paris, 1992). 3. D. Burghelea, L. Friedlander, T. Kappeler and P. McDonald, ‘Analytic and Reidemeister torsion for representations in ﬁnite type Hilbert modules’, Geom. Funct. Anal. 6 (1996) 751–859.