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Towards the definition of metric hyperbolicity. (English) Zbl 1109.37002
Summary: 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 [the author, Discrete Contin. Dyn. Syst. 13, 1305–1324 (2005; Zbl 1115.37002)] and St. Petersburg Math. J. 17, 763–772 (2006; Zbl 1173.47306)].
##### MSC:
 37A05 Dynamical aspects of measure-preserving transformations 60J05 Discrete-time Markov processes on general state spaces 37A25 Ergodicity, mixing, rates of mixing 47A35 Ergodic theory of linear operators 37A30 Ergodic theorems, spectral theory, Markov operators 28D99 Measure-theoretic ergodic theory