Abstract

While there is, up to homeomorphism, only one Cantor space, i.e. one zero-dimensional, perfect, compact, nonempty metric space, there are many measures on Cantor space which are not topologically equivalent. The clopen values set for a full, nonatomic measure μ is the countable dense subset {μ(U): U is clopen} of the unit interval. It is a topological invariant for the measure. For the class of good measures, it is a complete invariant. A full, nonatomic measure μ is good if whenever U, V are clopen sets with μ(U) < μ(V), there exists W a clopen subset of V such that μ(W) = μ(U). These measures have interesting dynamical properties. They are exactly the measures which arise from uniquely ergodic minimal systems on Cantor space. For some of them there is a unique generic measure-preserving homeomorphism. That is, within the Polish group of such homeomorphisms there is a dense, G δ conjugacy class. ©2004 American Mathematical Society.

Department(s)

Mathematics and Statistics

Keywords and Phrases

Cantor set; Generic conjugacy class; Measure on Cantor space; Ordered measure spaces; Rohlin property; Unique ergodicity

International Standard Serial Number (ISSN)

0002-9947

Document Type

Article - Journal

Document Version

Citation

File Type

text

Language(s)

English

Rights

© 2026 American Mathematical Society, All rights reserved.

Publication Date

01 Jul 2005

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