Towards the definition of metric hyperbolicity

We introduce measure-theoretic definitions of 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 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 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.