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

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

2002-11-21

arxiv.org/abs/math/0211343v3

Mathematics

Dynamical Systems

This replaces the April 2004 version: a gap was fixed in Lemma 6 (regarding order of poles) and the Axioms corrected and gener

Scientific paper

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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