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Mosc. Math. J., 2005, Volume 5, Number 3, Pages 721–737 (Mi mmj217)  

This article is cited in 1 scientific paper (total in 2 paper)

Towards the definition of metric hyperbolicity

A. M. Vershik

St. Petersburg Department of V. A. Steklov Institute of Mathematics, Russian Academy of Sciences

Abstract: 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.

Key words and phrases: Polymorphisms, Markov operator, hyperbolic structure, quasisimilarity.

DOI: https://doi.org/10.17323/1609-4514-2005-5-3-721-737

Full text: http://www.ams.org/.../abst5-3-2005.html
References: PDF file   HTML file

Bibliographic databases:

MSC: Primary 37A05, 47A45, 60J27; Secondary 37A25, 37H10, 47A40
Received: July 4, 2005
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Citation: A. M. Vershik, “Towards the definition of metric hyperbolicity”, Mosc. Math. J., 5:3 (2005), 721–737

Citation in format AMSBIB
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\by A.~M.~Vershik
\paper Towards the definition of metric hyperbolicity
\jour Mosc. Math.~J.
\yr 2005
\vol 5
\issue 3
\pages 721--737
\mathnet{http://mi.mathnet.ru/mmj217}
\crossref{https://doi.org/10.17323/1609-4514-2005-5-3-721-737}
\mathscinet{http://www.ams.org/mathscinet-getitem?mr=2241819}
\zmath{https://zbmath.org/?q=an:1109.37002}
\isi{http://gateway.isiknowledge.com/gateway/Gateway.cgi?GWVersion=2&SrcApp=PARTNER_APP&SrcAuth=LinksAMR&DestLinkType=FullRecord&DestApp=ALL_WOS&KeyUT=000208595500013}


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    Citing articles on Google Scholar: Russian citations, English citations
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    This publication is cited in the following articles:
    1. Thouvenot J.-P., “Two facts concerning the transformations which satisfy the weak Pinsker property”, Ergodic Theory and Dynamical Systems, 28:2 (2008), 689–695  crossref  mathscinet  zmath  isi  scopus
    2. V. M. Buchstaber, M. I. Gordin, I. A. Ibragimov, V. A. Kaimanovich, A. A. Kirillov, A. A. Lodkin, S. P. Novikov, A. Yu. Okounkov, G. I. Olshanski, F. V. Petrov, Ya. G. Sinai, L. D. Faddeev, S. V. Fomin, N. V. Tsilevich, Yu. V. Yakubovich, “Anatolii Moiseevich Vershik (on his 80th birthday)”, Russian Math. Surveys, 69:1 (2014), 165–179  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib
  • Moscow Mathematical Journal
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