Counting the relatively finite factors of a Bernoulli shift

Abstract

LetT acting on (Ω, ℱ, μ) be afinite entropy Bernoulli shift. AT invariant factor Open image in new window is “relatively finite” if a.e. fiber of\(\mathfrak{A}\) has a finite, hence constantk, number of points. We say two factors Open image in new window , “sit the same” if there is a measurable measure preserving map ϕ with\({}_\varphi T_\varphi ^{ - 1} = T\) and\({}_\varphi (\mathfrak{A}) = \mathfrak{A}'\). We show here that up to sitting the same there are only finitely many relatively finite factors withk point fibers in a Bernoulli shift, and that they are classified by a certain algebraic structure in the symmetric group onk-points.