Abstract

We study the structure of the chains of partitions ? of countable sets. Our main result asserts that for any chain ? and any probability measure ? the set of entropy values h?(?) = {h?(?) : ? ? ?} is a totally disconnected set in R+ U {+?} with null-Lebesgue measure. The complexity of the set h?(?) is exhibited in an example where h?(?) is a Cantor set with non null Hausdorff dimension.