  A Borel system consists of a measurable automorphism of a standard Borel
space. We consider Borel embeddings and isomorphisms between such systems
modulo null sets, i.e. sets which have measure zero for every invariant
probability measure. For every t>0 we show that in this category there exists a
unique free Borel system (Y,S) which is strictly t-universal in the sense that
all invariant measures on Y have entropy <t, and if (X,T) is another free
system obeying the same entropy condition then X embeds into Y off a null set.
One gets a strictly t-universal system from mixing shifts of finite type of
entropy at least t by removing the periodic points and "restricting" to the
part of the system of entropy <t. As a consequence, after removing their
periodic points the systems in the following classes are completely classified
by entropy up to Borel isomorphism off null sets: mixing shifts of finite type,
mixing positive-recurrent countable state Markov chains, mixing sofic shifts,
beta shifts, synchronized subshifts, and axiom-A diffeomorphisms. In particular
any two equal-entropy systems from these classes are entropy conjugate in the
sense of Buzzi, answering a question of Boyle, Buzzi and Gomez.
