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2013 | 11 | 3 | 379-400
Tytuł artykułu

Geometry and dynamics of admissible metrics in measure spaces

Treść / Zawartość
Warianty tytułu
Języki publikacji
EN
Abstrakty
EN
We study a wide class of metrics in a Lebesgue space, namely the class of so-called admissible metrics. We consider the cone of admissible metrics, introduce a special norm in it, prove compactness criteria, define the ɛ-entropy of a measure space with an admissible metric, etc. These notions and related results are applied to the theory of transformations with invariant measure; namely, we study the asymptotic properties of orbits in the cone of admissible metrics with respect to a given transformation or a group of transformations. The main result of this paper is a new discreteness criterion for the spectrum of an ergodic transformation: we prove that the spectrum is discrete if and only if the ɛ-entropy of the averages of some (and hence any) admissible metric over its trajectory is uniformly bounded.
Wydawca
Czasopismo
Rocznik
Tom
11
Numer
3
Strony
379-400
Opis fizyczny
Daty
wydano
2013-03-01
online
2012-12-22
Twórcy
  • St. Petersbrug Branch of Mathematical Institute of Russian Academy of Science, Fontanka 27, 191023, St. Petersbrug, Russa, avershik@gmail.com
  • St. Petersbrug Branch of Mathematical Institute of Russian Academy of Science, Fontanka 27, 191023, St. Petersbrug, Russa, paxa239@yandex.ru
autor
  • St. Petersbrug Branch of Mathematical Institute of Russian Academy of Science, Fontanka 27, 191023, St. Petersbrug, Russa, fedyapetrov@gmail.com
Bibliografia
  • [1] Feldman J., r-entropy, equipartition, and Ornstein’s isomorphism theorem in ℝn, Israel J. Math., 1980, 36(3–4), 321–345 http://dx.doi.org/10.1007/BF02762054[Crossref]
  • [2] Ferenczi S., Measure-theoretic complexity of ergodic systems, Israel J. Math., 1997, 100, 189–207 http://dx.doi.org/10.1007/BF02773640[Crossref]
  • [3] Ferenczi S., Park K.K., Entropy dimensions and a class of constructive examples, Discrete Contin. Dyn. Syst., 2007, 17(1), 133–141
  • [4] Gromov M., Metric Structures for Riemannian and Non-Riemannian Spaces, Progr. Math., 152, Birkhäuser, Boston, 1999
  • [5] Katok A., Thouvenot J.-P., Slow entropy type invariants and smooth realization of commuting measure-preserving transformation, Ann. Inst. H.Poincaré, 1997, 33(3), 323–338 http://dx.doi.org/10.1016/S0246-0203(97)80094-5[Crossref]
  • [6] Kushnirenko A.G., Metric invariants of entropy type, Russian Math. Surveys, 1967, 22(5), 53–61 http://dx.doi.org/10.1070/RM1967v022n05ABEH001225[Crossref]
  • [7] Ornstein D.S., Weiss B., Entropy and isomorphism theorems for actions of amenable groups, J. Analyse Math., 1987, 48, 1–141 http://dx.doi.org/10.1007/BF02790325[Crossref]
  • [8] Rokhlin V.A., On the fundamental ideas of measure theory, Mat. Sb., 1949, 25(67)(2), 107–150 (in Russian)
  • [9] Stromberg K., An elementary proof of Steinhaus’s theorem, Proc. Amer. Math. Soc., 1972, 36(1), 308
  • [10] Vershik A.M., The universal Uryson space, Gromov’s metric triples, and random metrics on the series of natural numbers, Russian Math. Surveys, 1998, 53(5), 921–928 http://dx.doi.org/10.1070/RM1998v053n05ABEH000069[Crossref]
  • [11] Vershik A.M., Dynamic theory of growth in groups: entropy, boundaries, examples, Russian Math. Surveys, 2000, 55(4), 667–733 http://dx.doi.org/10.1070/RM2000v055n04ABEH000314[Crossref]
  • [12] Vershik A.M., Classification of measurable functions of several arguments, and invariantly distributed random matrices, Funct. Anal. Appl., 2002, 36(2), 93–105 http://dx.doi.org/10.1023/A:1015662321953[Crossref]
  • [13] Vershik A.M., Random and universal metric spaces, In: Fundamental Mathematics Today, Independent University of Moscow, Moscow, 2003, 54–88 (in Russian)
  • [14] Vershik A.M., Random metric spaces and universality, Russian Math. Surveys, 2004, 59(2), 259–295 http://dx.doi.org/10.1070/RM2004v059n02ABEH000718[Crossref]
  • [15] Vershik A.M., Dynamics of metrics in measure spaces and their asymptotic invariant, Markov Process. Related Fields, 2010, 16(1), 169–184
  • [16] Vershik A.M., Information, entropy, dynamics, In: Mathematics of the 20th Century. A View from Petersburg, Moscow Center for Continuous Mathematical Education, Moscow, 2010, 47–76 (in Russian)
  • [17] Vershik A.M., Scaling entropy and automorphisms with pure point spectrum, St. Petersburg Math. J., 2012, 23(1), 75–91 http://dx.doi.org/10.1090/S1061-0022-2011-01187-2[Crossref]
  • [18] Vershik A.M., Gorbulsky A.D., Scaled entropy of filtrations of σ-fields, Theory Probab. Appl., 2008, 52(3), 493–508 http://dx.doi.org/10.1137/S0040585X97983122[Crossref]
  • [19] Zatitskiy P.B., Petrov F.V., Correction of metrics, J. Math. Sci., 2012, 181(6), 867–870 http://dx.doi.org/10.1007/s10958-012-0720-8[Crossref]
Typ dokumentu
Bibliografia
Identyfikatory
Identyfikator YADDA
bwmeta1.element.doi-10_2478_s11533-012-0149-9
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