Pointwise Assouad dimension for measures

TL;DR

This paper introduces a pointwise version of the Assouad dimension for measures, explores its properties, and compares it to the global Assouad dimension, revealing similarities in classical cases and computing dimensions for specific invariant measures.

Contribution

It defines and analyzes the pointwise Assouad dimension, establishing its relation to the global dimension and computing it for certain self-conformal measures.

Findings

Pointwise Assouad dimension often matches the global Assouad dimension in classical cases.

The pointwise Assouad dimension differs from the global one in general.

Computed the Assouad dimension for invariant measures on self-conformal sets.

Abstract

We introduce a pointwise variant of the Assouad dimension for measures on metric spaces, and study its properties in relation to the global Assouad dimension. We show that, in general, the value of the pointwise Assouad dimension differs from the global counterpart, but in many classical cases, it exhibits similar exact dimensionality properties as the classical local dimension, namely it equals the global Assouad dimension at almost every point. We also compute the Assouad dimension of invariant measures with place dependent probabilities supported on strongly separated self-conformal sets.