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Journal of Topology Advance Access originally published online on October 25, 2007
Journal of Topology 2008 1(1):115-151; doi:10.1112/jtopol/jtm005
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© 2007 London Mathematical Society

Dynamics, Laplace transform and spectral geometry

Dan Burghelea

Department of Mathematics, The Ohio State University, 231 West Avenue, Columbus, OH 43210, USA burghele{at}math.ohio-state.edu

Stefan Haller

Department of Mathematics, University of Vienna, Nordbergstraße 15, A-1090, Vienna, Austria stefan.haller{at}univie.ac.at


  Abstract

In this paper, we consider vector fields on a closed manifold whose instantons and closed trajectories can be ‘counted’. Vector fields which admit Lyapunov closed one forms belong to this class. We show that under an additional hypothesis, ‘the exponential growth property’, the counting functions of instantons and closed trajectories have Laplace transforms which can be related to the topology and the geometry of the underlying manifold. The purpose of this paper is to introduce and explore the concept ‘exponential growth property’, and to describe these Laplace transforms.

Received February 24, 2007.

2000 Mathematics Subject Classification 57R20, 57R58, 57R70, 57Q10, 58J52

Part of this work was done while the second author enjoyed the warm hospitality of the Ohio State University. The second author was partially supported by the Fonds zur Förderung der wissenschaftlichen Forschung (Austrian Science Fund), project number P14195-MAT. Part of this work was carried out when the first author enjoyed the hospitality of IHES in Bures sur Yvette and the second author had the pleasure of visiting the Max Planck Institute for Mathematics in Bonn.


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