Duke Mathematical Journal

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

Mathieu Baillif

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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]).

Article information

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

Dates
First available in Project Euclid: 30 July 2004

Permanent link to this document
https://projecteuclid.org/euclid.dmj/1091217477

Digital Object Identifier
doi:10.1215/S0012-7094-04-12415-7

Mathematical Reviews number (MathSciNet)
MR2072214

Zentralblatt MATH identifier
1330.37027

Subjects
Primary: 37C30: Zeta functions, (Ruelle-Frobenius) transfer operators, and other functional analytic techniques in dynamical systems

Citation

Baillif, Mathieu. Kneading operators, sharp determinants, and weighted Lefschetz zeta functions in higher dimensions. Duke Math. J. 124 (2004), no. 1, 145--175. doi:10.1215/S0012-7094-04-12415-7. https://projecteuclid.org/euclid.dmj/1091217477


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