We study the behaviour of points in the boundaries of periodic Fatou
components for transcendental entire of meromorphic maps. We show that for
a simply connected invariant Baker domain U of a meromorphic map f with a
finite degree on U, either a typical point in the boundary (with respect
to harmonic measure) escapes to infinity, or a typical point has a dense
trajectory in the boundary of U, depending on the Baker-Cowen-Pommerenke
type of U. If U is parabolic, only the second case occurs. Furthermore, if
U is a component of the basin of an attracting periodic orbit of period
larger than 1 for an exponential map, a typical point (in the sense of
dimension) does not tend to infinity. The talk is based on joint results
with N. Fagella, X. Jarque, B. Karpińska and A. Zdunik.