  Quantization dimension has been computed for many invariant measures of
dynamically defined fractals having well separated cylinders, that is, in the
cases when the so-called Open Set Condition (OSC) holds. To attack the same
problem in case of heavy overlaps between the cylinders, we consider a family
of self-similar measures, for which the underlying Iterated Function System
satisfies the so-called Weak Separation Property (WSP) but does not satisfy the
OSC since complete overlaps occur in between the cylinders. The work in this
paper also shows that the quantization dimension determined for the set of
overlap self-similar construction satisfying the WSP has a relationship with
the temperature function of the thermodynamic formalism.
