  For a hyperbolic map f on a saddle type fractal Lambda with
self-intersections, the number of f- preimages of a point x in Lambda may
depend on x. This makes estimates of the stable dimensions more difficult than
for diffeomorphisms or for maps which are constant-to-one. We employ the
thermodynamic formalism in order to derive estimates for the stable Hausdorff
dimension function delta^s on Lambda, in the case when f is conformal on local
stable manifolds. These estimates are in terms of a continuous function on
Lambda which bounds the preimage counting function from below. As a corollary
we obtain that if delta^s attains its maximal possible value in Lambda, then
the stable dimension is constant throughout Lambda, whereas the preimage
counting function is constant on at least an open and dense subset of Lambda.
In particular, this shows that if at some point in Lambda, the stable dimension
is equal to the analogue of the similarity dimension in the stable direction at
that point, then f behaves very much like a homeomorphism on Lambda. Finally we
also obtain results about the stable upper box dimension for these type of
fractals. We end the paper with a discussion of two explicit examples.
