Dynamical Determinants via Dynamical Conjugacies for Postcritically Finite Polynomials

Abstract

We give an analogue of Levin–Sodin–Yuditskii's study of the dynamical Ruelle determinants of hyperbolic rational maps in the case of subhyperbolic quadratic polynomials. Our main tool is to reduce to an expanding situation. We do so by applying a dynamical change of coordinates on the domains of a Markov partition constructed from the landing ray at the postcritical repelling orbit. We express the dynamical determinants $d_\beta (z) = \exp - \sum {_{k \geqslant 1} } \tfrac{{z^k }}{k}\sum {_{w \in {\text{Fix }}f^k } \tfrac{1}{{((f_c^k )'(w))^\beta }}\tfrac{1}{{1 - \tfrac{1}{{(f_c^k )'(w)}}}}(\beta \in \mathbb{Z}_ + )}$ as the product of an (entire) determinant with an explicit expression involving the postcritical repelling orbit, thus explaining the poles in d _β (z).