Uniform endomorphisms which are isomorphic to a Bernoulli shift
Abstract
A {\it uniformly
p
-to-one endomorphism} is a measure-preserving map with entropy log
p
which is almost everywhere
p
-to-one and for which the conditional expectation of each preimage is precisely
1/p
. The {\it standard} example of this is a one-sided
p
-shift with uniform i.i.d. Bernoulli measure. We give a characterization of those uniformly finite-to-one endomorphisms conjugate to this standard example by a condition on the past tree of names which is analogous to {\it very weakly Bernoulli} or {\it loosely Bernoulli.} As a consequence we show that a large class of isometric extensions of the standard example are conjugate to it.