Geometry and ergodic theory of non-hyperbolic exponential maps

We deal with all the maps from the exponential family such that the orbit of zero escapes to infinity sufficiently fast. In particular all the parameters are included. We introduce as our main technical devices the projection of the map to the infinite cylinder and an appropriate conformal measure . We prove that , essentially the set of points in returning infinitely often to a compact region of disjoint from the orbit of , has the Hausdorff dimension , that the -dimensional Hausdorff measure of is positive and finite, and that the -dimensional packing measure is locally infinite at each point of . We also prove the existence and uniqueness of a Borel probability -invariant ergodic measure equivalent to the conformal measure . As a byproduct of the main course of our considerations, we reprove the result obtained independently by Lyubich and Rees that the -limit set (under ) of Lebesgue almost every point in , coincides with the orbit of zero under the map . Finally we show that the the function , , is continuous.