# Geometry and dynamics of admissible metrics in measure spaces

Anatoly Vershik; Pavel Zatitskiy; Fedor Petrov

Open Mathematics (2013)

- Volume: 11, Issue: 3, page 379-400
- ISSN: 2391-5455

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topAnatoly Vershik, Pavel Zatitskiy, and Fedor Petrov. "Geometry and dynamics of admissible metrics in measure spaces." Open Mathematics 11.3 (2013): 379-400. <http://eudml.org/doc/269752>.

@article{AnatolyVershik2013,

abstract = {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.},

author = {Anatoly Vershik, Pavel Zatitskiy, Fedor Petrov},

journal = {Open Mathematics},

keywords = {Admissible metric; Measure space; Automophisms; Scaling entropy; Criteria of discreteness spectrum; admissible metric; measure space; measure-preserving transformation; scaling entropy; discrete spectrum},

language = {eng},

number = {3},

pages = {379-400},

title = {Geometry and dynamics of admissible metrics in measure spaces},

url = {http://eudml.org/doc/269752},

volume = {11},

year = {2013},

}

TY - JOUR

AU - Anatoly Vershik

AU - Pavel Zatitskiy

AU - Fedor Petrov

TI - Geometry and dynamics of admissible metrics in measure spaces

JO - Open Mathematics

PY - 2013

VL - 11

IS - 3

SP - 379

EP - 400

AB - 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.

LA - eng

KW - Admissible metric; Measure space; Automophisms; Scaling entropy; Criteria of discreteness spectrum; admissible metric; measure space; measure-preserving transformation; scaling entropy; discrete spectrum

UR - http://eudml.org/doc/269752

ER -

## References

top- [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] Zbl1095.28510
- [3] Ferenczi S., Park K.K., Entropy dimensions and a class of constructive examples, Discrete Contin. Dyn. Syst., 2007, 17(1), 133–141 Zbl1128.37004
- [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] Zbl0884.60009
- [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] Zbl0169.46101
- [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] Zbl0637.28015
- [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 Zbl0278.28013
- [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] Zbl1005.53036
- [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] Zbl0991.37005
- [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 Zbl1203.37020
- [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] Zbl1161.28005
- [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] Zbl1259.28018

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