  We introduce measure-theoretic definitions of {\it hyperbolic structure for
measure-preserving automorphisms}. A wide class of $K$-automorphisms possesses
a hyperbolic structure; we prove that all $K$-automorphisms have a slightly
weaker structure of {\it semi-hyperbolicity}. Instead of the notions of stable
and unstable foliations and other notions from smooth theory, we use the tools
of the theory of polymorphisms. The central role is played by {\it
polymorphisms} associated with a special invariant equivalence relation, more
exactly, with a homoclinic equivalence relation. We call an automorphism with
given hyperbolic structure a hyperbolic automorphism and prove that it is
canonically quasi-similar to a so-called prime nonmixing polymorphism.
  We present a short but necessary vocabulary of polymorphisms and Markov
operators from \cite{V1,V2}.
