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15 July 2004 Kneading operators, sharp determinants, and weighted Lefschetz zeta functions in higher dimensions
Mathieu Baillif
Duke Math. J. 124(1): 145-175 (15 July 2004). DOI: 10.1215/S0012-7094-04-12415-7

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: $\mathrm{Det}^{\#}(1-z\mathcal{M})=\Pi_{k=0}^n \mathrm{Det}^{\flat}(1-z\mathcal{M}^{(k)})^{(-1)^k}$. 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)

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

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]).

Citation

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Mathieu Baillif. "Kneading operators, sharp determinants, and weighted Lefschetz zeta functions in higher dimensions." Duke Math. J. 124 (1) 145 - 175, 15 July 2004. https://doi.org/10.1215/S0012-7094-04-12415-7

Information

Published: 15 July 2004
First available in Project Euclid: 30 July 2004

zbMATH: 1330.37027
MathSciNet: MR2072214
Digital Object Identifier: 10.1215/S0012-7094-04-12415-7

Subjects:
Primary: 37C30

Rights: Copyright © 2004 Duke University Press

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Vol.124 • No. 1 • 15 July 2004
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