Zeros of dynamical zeta functions for hyperbolic quadradic maps

TL;DR

This paper proves that the dynamical zeta function for certain quadratic maps has zero-free regions beyond the line Re(s)=1/2, and provides numerical plots of zeros.

Contribution

It establishes the existence of zero-free strips for the zeta function of quadratic maps with specific parameters, advancing understanding of their zero distribution.

Findings

Zero-free strips of size 1/2+ for the zeta function

Finite zeros in the region Re(s) > 1/2 + epsilon

Numerical plots illustrating zero distribution

Abstract

We prove that the dynamical zeta function $Z(s)$ associated to $z^2 + c$ with $c < -3.75$ has essential zero-free strips of size $1/2 +$, that is, for every $\epsilon > 0$, there exist only finitely many zeros in the strip $\mathrm{Re}(s) > 1/2 + \epsilon$. We also present some numerical plots of zeros of $Z(s)$ using the method proposed in Jenkinson-Pollicott.