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Kneading determinants and spectra of transfer operators in higher dimensions, the isotropic case

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Published in Ergodic Theory and Dynamical Systems. 2005, vol. 25, no. 5, p. 1437-1470
Abstract Transfer operators M_k acting on k-forms in R^n are associated to smooth transversal local diffeomorphisms and compactly supported weight functions. A formal trace is defined by summing the product of the weight and the Lefschetz sign over all fixed points of all the diffeos. This yields a formal Ruelle-Lefschetz determinant Det^#(1-zM). We use the Milnor-Ruelle-Kitaev equality (recently proved by Baillif), which expressed Det^#(1-zM) as an alternated product of determinants of kneading operators,Det(1+D_k(z)), to relate zeroes and poles of the Ruelle-Lefschetz determinant to the spectra of the transfer operators M_k. As an application, we get a new proof of a theorem of Ruelle on smooth expanding dynamics.
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BAILLIF, Mathieu, BALADI, Viviane. Kneading determinants and spectra of transfer operators in higher dimensions, the isotropic case. In: Ergodic Theory and Dynamical Systems, 2005, vol. 25, n° 5, p. 1437-1470. https://archive-ouverte.unige.ch/unige:12219

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Deposited on : 2010-10-22

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