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

## TL;DR

This paper extends multifractal analysis of finite type self-similar measures from equicontractive to non-equicontractive cases, broadening applicability and including measures that fail the open set condition.

## Contribution

It generalizes the concept of regularity to generalized regularity for non-equicontractive measures, preserving key properties and results from the equicontractive setting.

## Key findings

- Many results from equicontractive case carry over to non-equicontractive setting.
- Introduction of generalized regularity includes non-equicontractive measures.
- Examples demonstrate the applicability of the generalized regularity concept.

## Abstract

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 {\em regular}, then the calculations of local dimensions were relatively easy. We modify this definition of regular to define measures to be {\em generalized regular}. This new definition will include the non-equicontractive case and obtain similar results. Examples are studied of non-equicontractive self-similar generalized regular measures, as well as equicontractive self-similar measures which generalized regular in this new sense, but which are not regular.