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Since λ ∈ Lη , we have λ η < ∞. Using the Novikov property of λ and the injectivity of ω : Γ → R again, we ﬁnd t0 ∈ R so that 1 − λ η +tω < 1 for all t t0 . Since L1η +tω is a Banach algebra, k 0 (1 − λ)k will converge and λ−1 ∈ Λω ∩ L1η +tω for all t t0 . + Recall that we have a bijection exp : Λ+ ω → 1 + Λω . 3. Suppose λ ∈ Λ+ ω and exp(λ) ∈ Lη . Then, there exists t0 ∈ R so that λ ∈ 1 Λω ∩ Lη +tω for all t t0 . Proof. 2 using log(1 − µ) = − µk k>0 k . 6. 6) in terms of closed trajectories under some assumptions.
P. Novikov, ‘Quasiperiodic structures in topology’, Topological methods in modern mathematics, Proceedings of the Symposium in Honor of John Milnor’s Sixtieth Birthday, Stony Brook, NY, 1991 (ed. L. R. Goldberg and A. V. Phillips; Publish or Perish, Houston, TX, 1993) 223–233. 18. A. V. Pajitnov, ‘The incidence coeﬃcients in the Novikov complex are generically rational functions’, St. Petersburg Math. J. 9 (1998) 969–1006; translation from Algebra i Analiz 9 (1997) 92–139. 19. A. V. Pajitnov, ‘C 0 -generic properties of boundary operators in the Novikov complex’, Pseudoperiodic topology, American Mathematical Society Translations 197 (American Mathematical Society, Providence, RI, 1999) 29–115.
D. Burghelea and S. Haller, ‘Torsion as a function on the space of representations’, Proceedings of the a user, Basel), to appear. Conference on C ∗ -algebra and elliptic theory II, Trends in Mathematics (Birkh¨ 7. D. Fried, ‘Lefschetz formulas for ﬂows’, Proceedings of the Lefschetz Centennial Conference, Part III, Mexico City, Mexico, 1984, Contemporary Mathematics 58 (American Mathematical Society, Providence, RI, 1987) 19–69. 8. E. Hille and R. S. Phillips, Functional analysis and semi-groups, Third printing of the revised edition of 1957, American Mathematical Society Colloquium Publications XXXI (American Mathematical Society, Providence, RI, 1974).
basicelements of differential geometry and topology