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

M. Baillif·V. Baladi
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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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M. Baillif, V. Baladi. 2004-06-21. Kneading determinants and spectra of transfer operators in higher dimensions, the isotropic case. https://arxiv.org/abs/math/0211343

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