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

2002/11/21 by M. Baillif, Baillif, M., V. Baladi +1
Mathematics · #37C30 #Dynamical Systems (math.DS) #FOS: Mathematics #Functional Analysis (math.FA) #math.DS #math.FA #msc:37C30

paper · pdf · doi:10.48550/arxiv.math/0211343

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

arxiv created 2004/06/21 · arxiv updated 2009/11/30

Abstract

Transfer operators M_k acting on k-forms in Rⁿ 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.

Related

• Dynamical determinants and spectrum for hyperbolic diffeomorphisms
• Dynamical zeta functions
• Spectra of differentiable hyperbolic maps
• Fredholm determinants, Anosov maps and Ruelle resonances
• The role of transfer operators and shifts in the study of fractals: encoding-models, analysis and geometry, commutative and non-commutative