# Uniform endomorphisms which are isomorphic to a Bernoulli shift

### Abstract

A uniformly $p$-toone endomorphism is a measure-preserving map with entropy log $p$ which is almost everywhere $p$-toone and for which the conditional expectation of each preimage is precisely $1/p$. The 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 three of names which is analogous to very weak Bernoulli or loosely Bernoulli. As a consequence we show that a large class of isometric extensions of the standard example are conjugate to it.

## Authors

Christopher Hoffman

Daniel Rudolph