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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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References

REFERENCES

[1]Anosov, D. V.. Tangent fields of transversal foliations in ‘U-systems’. Math. Zametki 2 (5) (1967), 539–548. (Russian)Google Scholar
[2]Craig, W. & Simon, B.. Subharmonicity of the Lyapunov index. Duke Math. J. 50 (2) (1983), 551–560.CrossRefGoogle Scholar
[3]Deift, P. & Simon, B.. Almost periodic Schrödinger operators III. The absolutely continuous spectrum in one-dimension. Commun. Math. Phys. 90 (1983), 389–411.CrossRefGoogle Scholar
[4]Dubrovin, B., Matveev, V. & Novikov, S.. Nonlinear equations of Korteweg- de Vries type, finite zone linear operators, and Abelian varieties. Russ. Math. Surv. 31 (1976), 59–146.CrossRefGoogle Scholar
[5]Eberlein, P.. When is a geodesic flow of Anosov type? I. J. Diff. Geom. 8 (1973), 437–463.Google Scholar
[6]Guillemin, V. & Duistermaat, J.. The spectrum of a positive elliptic operator and periodic bicharacteristics. Invent. Math. 29 (1975), 37–79.Google Scholar
[7]Johnson, R. & Moser, J.. The rotation number of almost periodic potentials, Commun. Math. Phys. 84 (1982), 403–438.CrossRefGoogle Scholar
[8]Katok, A.. Entropy and closed geodesics. Ergod. Th. & Dynam. Sys. 2 (1982), 339.CrossRefGoogle Scholar
[9]Kotani, S.. Ljapounov indices determine absolutely continuous spectra of stationary random one-dimensional Schrödinger operators. Proc. Taniguchi Symp. S. A. Katata (1982), pp. 225–247.Google Scholar
[10]Kotani, S. & Simon, B.. Stochastic Schrödinger operators and Jacobi matrices on the Strip. Commun. Math. Phys. 119 (1988), 403–429.CrossRefGoogle Scholar
[11]Manning, A.. Topological entropy for geodesic flows. Ann. Math. 110 (1979), 567–573.CrossRefGoogle Scholar
[12]McKean, H. & van Moerbeke, P.. The spectrum of Hill's equation. Invent. Math. 30 (1975), 217–274.CrossRefGoogle Scholar
[13]McKean, H. & Trubowitz, E.. Hill's operator and hyperelliptic function theory in the presence of infinitely many branch points. Commun. Pure Appl. Math. 29 (1976), 153–226.CrossRefGoogle Scholar
[14]Moser, J.. An example of a Schrödinger operator with almost periodic potential and nowhere dense spectrum. Comm. Math. Helv. 56 (1981), 198–224.CrossRefGoogle Scholar
[15]Osserman, R. & Sarnak, P.. A new curvature invariant and entropy of geodesic flows. Invent. Math. 77 (1984), 455–462.CrossRefGoogle Scholar
[16]Pesin, Ya. B.. Characteristic Lyapunov exponents and smooth ergodic theory. Russ. Math. Surv. 4 (32) (1977), 55–114.CrossRefGoogle Scholar