Automorphism groups of measures on the Cantor space. Part II: Abstract homogeneous measures

TL;DR

This paper introduces a new class of ultrahomogeneous measures on the Cantor space, characterizes their automorphism groups, and explores conditions for minimality and dense conjugacy classes, advancing understanding of measure symmetries.

Contribution

It defines and studies a novel class of ultrahomogeneous measures on the Cantor space, characterizes their automorphism groups, and links minimal homeomorphisms to universal invariant measures.

Findings

Existence of uncountably many non-atomic ultrahomogeneous measures with dense automorphism group conjugacy classes.

Characterization of measures with transitive or minimal automorphism group actions via trinary spectrum.

Any minimal homeomorphism induces a universal h-invariant measure, unique up to Q-linear isomorphism.

Abstract

The main aim of the paper is to introduce a new class of (semigroup-valued) measures that are ultrahomogeneous on the Boolean algebra of all clopen subsets of the Cantor space and to study their automorphism groups. A characterisation, in terms of the so-called trinary spectrum of a measure, of ultrahomogeneous measures such that the action of their automorphism groups is (topologically) transitive or minimal is given. Also sufficient and necessary conditions for the existence of a dense (or co-meager) conjugacy class in these groups are offered. In particular, it is shown that there are uncountably many full non-atomic probability Borel measures m on the Cantor space such that m and all its restrictions to arbitrary non-empty clopen sets have all the following properties: this measure is ultrahomogeneous and not good, the action of its automorphism group G is minimal (on a respective clopen set), and G has a dense conjugacy class. It is also shown that any minimal homeomorphism h on the Cantor space induces a homogeneous h-invariant probability measure that is universal among all h-invariant probability measures (which means that it `generates' all such measures) and this property determines this measure (in a certain sense) up to a Q-linear isomorphism.