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Homoclinic processes and invariant measures for hyperbolic toral automorphisms

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For every hyperbolic toral automorphism T, the present author has defined in his previous paper some unbounded T-invariant second-order difference operators related to the so-called homoclinic group of T. These operators were considered in the space L2 with respect to the Haar measure. It is shown in the present paper that such operators give rise to transition semigroups in the space of continuous functions on the torus and generate dynamically invariant Markov processes. This leads almost immediately to a family of invariant measures for the automorphism T.Along with a short discussion, some open questions about properties of these measures are posed. Bibliography: 9 titles.

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Authors and Affiliations

  1. St.Petersburg Department of the Steklov Mathematical Institute, St.Petersburg, Russia

    M. I. Gordin

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  1. M. I. Gordin
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Correspondence to M. I. Gordin.

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Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 368, 2009, pp. 122–129.

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Gordin, M.I. Homoclinic processes and invariant measures for hyperbolic toral automorphisms. J Math Sci 167, 501–505 (2010). https://doi.org/10.1007/s10958-010-9936-7

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  • DOI: https://doi.org/10.1007/s10958-010-9936-7

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