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

Abstract

Transfer operators ${\mathcal M}_k$ are associated to C^r transversal local diffeomorphisms $\psi_\omega$ of ${\mathbb R}^n$, and C^r compactly supported functions $g_\omega$. A formal trace $\operatorname{tr}^\# {\mathcal M}$, yields a formal Ruelle–Lefschetz determinant $\operatorname{Det}^\#(\operatorname{Id} -z{\mathcal M})$. We use the Milnor–Thurston–Kitaev equality recently proved by Baillif to relate zeros and poles of $\operatorname{Det}^\#(\operatorname{Id} -z{\mathcal M})$ with spectra of the transfer operators ${\mathcal M}_k$, under additional assumptions. As an application, we obtain a new proof of a result of Ruelle on the spectral interpretation of zeros and poles of the dynamical zeta function $\exp \sum_{m\ge1}(z^m/m) \sum_{f^m (x)=x} |{\rm det}\, Df(x)|^{-1}$ for smooth expanding endomorphisms f.