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

Abstract

We study transfer operators $\mathcal{M}^{(k)}$ associated to a finite family {ψ_ω} of $\mathcal{C}^{r}$ (r ≥ 1) transversal maps U_ω → $\mathbb{R}^n$, where U_ω ⊂ $\mathbb{R}^n$, with $\mathcal{C}^{r}$ compactly supported weights g_ω, 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]).