Kneading determinants and spectra of transfer operators in higher dimensions, the isotropic case
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.
Keywords
Cite
@article{arxiv.math/0211343,
title = {Kneading determinants and spectra of transfer operators in higher dimensions, the isotropic case},
author = {M. Baillif and V. Baladi},
journal= {arXiv preprint arXiv:math/0211343},
year = {2007}
}
Comments
This replaces the April 2004 version: a gap was fixed in Lemma 6 (regarding order of poles) and the Axioms corrected and generalised