One of the questions of ergodic
theory is that of “relative position” of factors of Bernoulli shift. If ℱ0 and ℱ1 are
factor algebras for a Bernoulli shift T, under what conditions is there an isomorphism
ϕ commuting with T such that ϕℱ0= ℱ1?
In this paper, we give an example of a Bernoulli shift T of a space X and
uncountably many partitions {Qα: α ∈ A} of X with the properties:
(T,Qα}≅(T,Qβ) for α,β ∈ A.
∨−∞∞TiQα is maximal for its entropy whenever α ∈ A.
There is no isomorphism ϕ commuting with T such that ϕQα= Qβ unless
α = β.
