Measures on Cantor sets: the good, the ugly, the bad

TL;DR

This paper explores the concept of 'good' measures on Cantor sets through the lens of traces on dimension groups, providing characterizations, examples, and connections to unperforation and convex geometry.

Contribution

It translates Akin's notions of goodness from Cantor measures to traces on dimension groups, offering new characterizations and constructing paradoxical examples.

Findings

Good traces are characterized by their kernels' dense image in affine functions.

Partial characterizations of refinability relate to convex subsets of Choquet simplices.

Examples illustrate the connection between goodness, refinability, and unperforation.

Abstract

We translate Akin's notion of {\it good} (and related concepts) from measures on Cantor sets to traces on dimension groups, and particularly for invariant measures of minimal homeomorphisms (and their corresponding simple dimension groups), this yields characterizations and examples, which translate back to the original context. Good traces on a simple dimension group are characterized by their kernel having dense image in their annihilating set of affine functions on the trace space; this makes it possible to construct many examples with seemingly paradoxical properties. In order to study the related property of {\it refinability,} we consider goodness for sets of measures (traces on dimension groups), and obtain partial characterizations in terms of (special) convex subsets of Choquet simplices. These notions also very closely related to unperforation of quotients of dimension groups by convex subgroups (that are not order ideals), and we give partial characterizations. Numerous examples illustrate the results.