We consider the one-dimensional classical Ising model in a symmetric dichotomous random field. The problem is reduced to a random iterated function system (RIFS) for an effective field. The D_q-spectrum of the invariant measure of this effective field exhibits a sharp drop of all D_q with q<0 at some critical strength of the random field. We introduce the concept of orbits, which naturally group the points of the support of the invariant measure. We then show that the pointwise dimension at all points of an orbit has the same value and calculate it for a class of periodic orbits and their so-called offshoots as well as for generic orbits in the non-overlapping case. The sharp drop in the D_q-spectrum is analytically explained by a drastic change of the scaling properties of the measure near the points of a certain periodic orbit at a critical strength of the random field, which is explicitly given. A similar drastic change near the points of a special family of periodic orbits explains a second, hitherto unnoticed transition in the D_q-spectrum. As it turns out, a decisive role in this mechanism is played by a specific offshoot. We furthermore give rigorous upper and/or lower bounds on all D_q in a wide parameter range. In most cases the numerically obtained D_q coincide with either the upper or the lower bound. The results in this paper are relevant for the understanding of RIFSs in the case of moderate overlap, in which periodic orbits with weak singularity can play a decisive role.