Attainable Assouad-like dimensions of randomly generated Moran sets and measures
Kathryn E. Hare, Franklin Mendivil
TL;DR
The paper analyzes Attainable Assouad-like Phi-dimensions for random Moran sets and measures generated by random weighted IFS with strong separation. It derives almost-sure formulas for the random set’s dimensions and characterizes the attainable range of Phi-dimensions for the measures under two regimes: dependent and independent probabilities. In the dependent case, any target upper Phi-dimension above the set-dimension threshold can be realized for the measures, while in the independent case explicit formulas yield a gap between attainable measure dimensions and the set's dimension, with Delta and delta governing reachability. The work further provides a precise framework for the interplay between local geometric behavior and randomness in one-variable fractals, along with extensions to more general Moran constructions and higher-dimensional settings.
Abstract
In this paper we study the Assouad-like $Φ$ dimensions of sets and measures that are constructed by a random weighted iterated function system of similarities. These dimensions are distinguished by the depth of the scales considered and thus provide more refined infomation about the local geometry/behaviour of a set or measure. The Assouad dimensions are important well-known examples. We determine the almost sure value of the upper and lower Assouad dimension of the random set. We also determine the range of attainable upper and lower (small) $Φ$ dimensions of the measures, in both the situation where the probability weights can depend on the scaling factors and when they cannot. In the later case we find that there is a ``gap'' between the dimension of the set and the dimensions of the associated family of random measures.