Geometry and ergodic theory of non-hyperbolic exponential maps

Abstract:

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.