Local dimensions of measures of finite type III - Measures that are not equicontractive

We extend the study of the multifractal analysis of the class of equicontractive self-similar measures of finite type to the non-equicontractive setting. Although stronger than the weak separation condition, the finite type property includes examples of IFS that fail the open set condition. The important combinatorial properties of equicontractive self-similar measures of finite type are extended to the non-equicontractive setting and we prove that many of the results from the equicontractive case carry over to this new, more general, setting. In particular, previously it was shown that if an equicontractive self-similar measure of finite type was regular, then the calculations of local dimensions were relatively easy. We modify this definition of regular to define measures to be generalized regular. This new definition will include the non-equicontractive case and we obtain similar results. Examples are studied of non-equicontractive self-similar generalized regular measures, as well as equicontractive self-similar measures which are generalized regular in this new sense, but which are not regular.