Uniform endomorphisms which are isomorphic to a Bernoulli shift

Details Uniform endomorphisms which are isomorphic to a Bernoulli shift Uniform endomorphisms which are isomorphic to a Bernoulli shift

2004-11-22

arxiv.org/abs/math/0411494v1

Ann. of Math. (2), Vol. 156 (2002), no. 1, 79--101

Mathematics

Dynamical Systems

23 pages, published version

Scientific paper

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.

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