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Duke Mathematical Journal

Kneading operators, sharp determinants, and weighted Lefschetz zeta functions in higher dimensions

Mathieu Baillif

Source: Duke Math. J. Volume 124, Number 1 (2004), 145-175.

Abstract

We study transfer operators $\mathcal{M}^{(k)}$ associated to a finite family {ψω} of $\mathcal{C}^{r}$ (r ≥ 1) transversal maps Uω → $\mathbb{R}^n, where Uω ⊂ $\mathbb{R}^n, with $\mathcal{C}^{r}$ compactly supported weights gω, acting on k-forms in $\mathbb{R}^n. Using the definitions of sharp trace Tr≯ and flat trace Tr≭, the following formula holds between power series: $\Det^{\#}(1-z\mathcal{M})=\Pi_{k=0}^n \Det^{\flat}(1-z\mathcal{M}^{(k)})^{(-1)^k}$. Following ideas of Kitaev [17], we define kneading operators $\mathcal{D}_k$(z), which are kernel operators. Our main result is the equality (as formal power series)

$$ \Det^{\#}(1-z\mathcal{M})=\prod_{k=0}^{n-1} \Det^{\flat}\big(1+\mathcal{D}_k(z)\big)^{(-1)^{k+1}}.$$

We also show that a finite power of $\mathcal{D}_k$(z) is trace-class on L2. This (partially) generalizes results obtained by Baladi, Kitaev, Ruelle, and Semmes in dimension one, complex and real [8], [10]).

Primary Subjects: 37C30

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Links and Identifiers

Permanent link to this document: http://projecteuclid.org/euclid.dmj/1091217477
Digital Object Identifier: doi:10.1215/S0012-7094-04-12415-7
Mathematical Reviews number (MathSciNet): MR2072214
Zentralblatt MATH identifier: 02103760

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