Local dimensions of measures of finite type II - Measures without full support and with non-regular probabilities

TL;DR

This paper extends the multifractal analysis of finite type self-similar measures to cases with non-regular probabilities and supports that are not intervals, identifying key classes where local dimensions behave predictably.

Contribution

It introduces the concept of a truly essential class for such measures and provides criteria for absolute continuity and support dimension computation.

Findings

Existence of a full-support subset with a closed interval of local dimensions

Presence of points with local dimension equal to the support's dimension within the essential class

Examples demonstrating isolated points in local dimension sets for specific measures

Abstract

Consider a sequence of linear contractions $S_{j}(x)=\varrho x+d_{j}$ and probabilities $p_{j}>0$ with $\sum p_{j}=1$. We are interested in the self-similar measure $\mu =\sum p_{j}\mu \circ S_{j}^{-1}$, of finite type. In this paper we study the multi-fractal analysis of such measures, extending the theory to measures arising from non-regular probabilities and whose support is not necessarily an interval. Under some mild technical assumptions, we prove that there exists a subset of supp$\mu $ of full $\mu $ and Hausdorff measure, called the truly essential class, for which the set of (upper or lower) local dimensions is a closed interval. Within the truly essential class we show that there exists a point with local dimension exactly equal to the dimension of the support. We give an example where the set of local dimensions is a two element set, with all the elements of the truly essential class giving the same local dimension. We give general criteria for these measures to be absolutely continuous with respect to the associated Hausdorff measure of their support and we show that the dimension of the support can be computed using only information about the essential class. To conclude, we present a detailed study of three examples. First, we show that the set of local dimensions of the biased Bernoulli convolution with contraction ratio the inverse of a simple Pisot number always admits an isolated point. We give a precise description of the essential class of a generalized Cantor set of finite type. Lastly, we study a maximal loop class that is not truly essential.