Iteration of closed geodesics in stationary Lorentzian manifolds |
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Authors: | Miguel Angel Javaloyes Levi Lopes de Lima Paolo Piccione |
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Affiliation: | 1. Departamento de Matemática, Universidade de S?o Paulo, Rua do Mat?o 1010, CEP 05508-900, S?o Paulo, SP, Brazil 2. Departamento de Matemática, Universidade Federal do Ceará, Campus do Pici, R. Humberto Monte, s/n, CEP 60455-760, Fortaleza, CE, Brazil
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Abstract: | Following the lines of Bott in (Commun Pure Appl Math 9:171–206, 1956), we study the Morse index of the iterates of a closed geodesic in stationary Lorentzian manifolds, or, more generally, of a closed Lorentzian geodesic that admits a timelike periodic Jacobi field. Given one such closed geodesic γ, we prove the existence of a locally constant integer valued map Λγ on the unit circle with the property that the Morse index of the iterated γ N is equal, up to a correction term εγ∈{0,1}, to the sum of the values of Λγ at the N-th roots of unity. The discontinuities of Λγ occur at a finite number of points of the unit circle, that are special eigenvalues of the linearized Poincaré map of γ. We discuss some applications of the theory. |
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Keywords: | KeywordHeading" >Mathematics Subject Classification (2000) 53C22 58E10 53C50 37B30 |
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