Dynamical Zeta functions for differentiable parabolic maps of the interval
Claudio Bonanno, Roberto Castorrini
TL;DR
The paper addresses the meromorphic extension of the dynamical zeta function $\zeta_{T,v}$ for differentiable parabolic maps of the interval with an indifferent fixed point, by inducing on the expanding part to form the jump map $G$ and the induced transfer operator $\mathcal{Q}_w(z)$. It establishes a meromorphic extension of $\zeta_{T,v}$ to the region $\{ |z|<1,\ \Lambda(z)<\rho^{r-2}\}$ with poles at $z_0=e^{-v(0)}$ and at eigenvalues of $\mathcal{Q}_w(z)$ (and zeros tied to $\mathcal{Q}_{w-\log G'}(z)$), and it shows that, under $v(0)<0$, $\mathcal{Q}_w(z)$ admits a larger analytic extension beyond the unit disc. Spectral analysis of the transfer operators reveals a Lasota–Yorke type framework for $G$ on $C^k([0,1])$ and highlights obstructions in the $C^1$-parabolic case, where the parabolic block can fail to produce a spectral gap. By linking the zeta function to flat determinants of these operators, the work generalizes dynamical zeta theory to differentiable parabolic maps and provides a pathway to broader analytic extensions with potential implications for statistical properties and phase transitions in non-uniformly expanding one-dimensional dynamics.
Abstract
This paper explores the domain of meromorphic extension for the dynamical zeta function associated to a class of one-dimensional differentiable parabolic maps featuring an indifferent fixed point. We establish the connection between this domain and the spectrum of the weighted transfer operators of the induced map. Furthermore, we discuss scenarios where meromorphic extensions occur beyond the confines of the natural disc of convergence of the dynamical zeta function.