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Floquet exponents for Jacobi fields

Published online by Cambridge University Press:  19 September 2008

Walter Craig
Walter Craig
Affiliation:
Department of Mathematics, Brown University, Providence, Rhode Island 02912, USA
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Abstract

This paper introduces a Riemannian invariant of a compact Riemannian manifold based on the spectral theory for the Jacobi field operator. It is the Floquet exponent for this operator, a purely dynamical quantity computable directly from the asymptotic behavior of Jacobi fields. We show that it is related to certain traces of the Green's function, and we derive further regularity and analyticity properties for the Green's function. In case the geodesic flow is ergodic, the Floquet exponent generalizes the measure entropy, and several entropy estimates follow. An asymptotic expansion of the Floquet exponent gives rise to a sequence of ‘Jacobi invariants’, which are related to the polynomial invariants of the K dV equation.

Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 11 , Issue 1 , March 1991 , pp. 41 - 63
Copyright
Copyright © Cambridge University Press 1991

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