Two Measures on Cantor Sets

TL;DR

This paper presents an example of a Cantor set where the equilibrium measure and Hausdorff measure are mutually absolutely continuous and both are regular in the Stahl-Totik sense.

Contribution

It provides a specific Cantor set example demonstrating mutual absolute continuity and regularity of equilibrium and Hausdorff measures, which was not previously known.

Findings

Equilibrium measure and Hausdorff measure are mutually absolutely continuous on the constructed Cantor set.

Both measures are regular in the Stahl-Totik sense.

The example illustrates new measure-theoretic properties of Cantor sets.

Abstract

We give an example of Cantor type set for which its equilibrium measure and the corresponding Hausdorff measure are mutually absolutely continuous. Also we show that these two measures are regular in Stahl-Totik sense.