Zeros of dynamical zeta functions for hyperbolic quadradic maps

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.