  We study the dimension of self-similar measures associated to a homogeneous
iterated function system of three contracting similarities on $\bf R$ and other
more general IFS's. We extend some of the theory recently developed for
Bernoulli convolutions to this setting. In the setting of three maps a new
phenomenon occurs, which has been highlighted by recent examples of Baker, and
B\'ar\'any, K\"aenm\"aki. To overcome the difficulties stemming form these, we
develop novel techniques, including an extension of Hochman's entropy increase
method to a function field setup.
