Measures with predetermined regularity and inhomogeneous self-similar sets

TL;DR

This paper investigates measures on inhomogeneous self-similar sets, establishing relationships between their dimensions and measures under certain conditions, revealing surprising differences between upper and lower regularity dimensions.

Contribution

It provides new results on the regularity and dimensional properties of measures supported on inhomogeneous self-similar sets, especially under the condensation open set condition.

Findings

Existence of measures with lower regularity dimension close to the set’s lower dimension.

Lower dimension of inhomogeneous self-similar set equals that of the condensation set under certain conditions.

Upper regularity dimension exceeds the maximum of the constituent sets' Assouad dimensions when the condensation set has smaller Assouad dimension.

Abstract

We show that if $X$ is a uniformly perfect complete metric space satisfying the finite doubling property, then there exists a fully supported measure with lower regularity dimension as close to the lower dimension of $X$ as we wish. Furthermore, we show that, under the condensation open set condition, the lower dimension of an inhomogeneous self-similar set $E_C$ coincides with the lower dimension of the condensation set $C$, while the Assouad dimension of $E_C$ is the maximum of the Assouad dimensions of the corresponding self-similar set $E$ and the condensation set $C$. If the Assouad dimension of $C$ is strictly smaller than the Assouad dimension of $E$, then the upper regularity dimension of any measure supported on $E_C$ is strictly larger than the Assouad dimension of $E_C$. Surprisingly, the corresponding statement for the lower regularity dimension fails.