Summary: Refinable Measures and Almost Unique Ergodicity
It is now known that the measures on Cantor space which arise from some
minimal, uniquely ergodic homeomorphism are characterized by a simple homo-
geneity condition referred to as `goodness', which for the two-symbol Bernoulli
trial Ár is equivalent to the condition that 1/r and 1/(1 - r) are algebraic in-
tegers, and that r is has exactly one algebraic conjugate (itself) lying in the
unit interval. One conjecture is that in the above statement, we may replace
`exactly one algebraic conjugate' with `exactly k algebraic conjugates', to ob-
tain a characterization of those measures which arise as the ergodic measures
for some minimal homeomorphism of Cantor space with exactly k ergodic mea-
sures. Such measures satisfy a slightly weaker homogeneity condition than good
measures, and are referred to as refinable.
In this talk, we give some support for the above conjecture by showing that
under certain circumstances, a refinable measure is equivalent to the product of
itself with a good measure, and so certain skew products are likely candidates
to be a minimal homeomorphism exactly k ergodic measures.