Geometry and ergodic theory of non-hyperbolic exponential maps

We deal with all the maps from the exponential family $\{\lambda e^z\}$ such that the orbit of zero escapes to infinity sufficiently fast. In particular all the parameters $\lambda\in (1/e,+\infty)$ are included. We introduce as our main technical devices the projection $F_{\lambda}$ of the map $f_{\lambda}$ to the infinite cylinder $Q=\mathbb{C}/2\pi i\mathbb{Z}$ and an appropriate conformal measure $m$. We prove that $J_r(F_\lambda)$, essentially the set of points in $Q$ returning infinitely often to a compact region of $Q$ disjoint from the orbit of $0\in Q$, has the Hausdorff dimension $h_\lambda\in (1,2)$, that the $h_\lambda$-dimensional Hausdorff measure of $J_r(F_\lambda)$ is positive and finite, and that the $h_\lambda$-dimensional packing measure is locally infinite at each point of $J_r(F_\lambda)$. We also prove the existence and uniqueness of a Borel probability $F_\lambda$-invariant ergodic measure equivalent to the conformal measure $m$. As a byproduct of the main course of our considerations, we reprove the result obtained independently by Lyubich and Rees that the $\omega$-limit set (under $f_\lambda$) of Lebesgue almost every point in $\mathbb{C}$, coincides with the orbit of zero under the map $f_\lambda$. Finally we show that the the function $\lambda\mapsto h_\lambda$, $\lambda\in (1/e,+\infty)$, is continuous.