  This thesis extends the Hoffman-Rudolph tvwB criterion defined for uniformly
p-to-1 endomorphisms to show that it still gives a sufficient condition for a
given $\bf p$-endomorphism to be one-sided Bernoulli. We also prove that given
any two tvwB group extensions of the one-sided $\bf p$-shift, we can find an
isomorphism preserving the natrual $\mathbb N \times G$-actions.
