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Mat. Sb., 2010, Volume 201, Number 7, Pages 3–14 (Mi msb7647)  

This article is cited in 7 scientific papers (total in 8 papers)

Extension of zero-dimensional hyperbolic sets to locally maximal ones

D. V. Anosov

Steklov Mathematical Institute, Russian Academy of Sciences

Abstract: It is proved that in any neighbourhood of a zero-dimensional hyperbolic set $F$ (hyperbolic sets are assumed to be compact) there is a locally maximal set $F_1\supset F$. In the proof, several already known or simple results are used, whose statements are given as separate assertions. The main theorem is compared with known related results, whose statements are also presented. (For example, it is known that the existence of $F_1$ is not guaranteed for $F$ of positive dimension.)
Bibliography: 7 titles.

Keywords: hyperbolic set, locally maximal, zero-dimensional, shadowing of pseudotrajectories and their families.

DOI: https://doi.org/10.4213/sm7647

Full text: PDF file (479 kB)
References: PDF file   HTML file

English version:
Sbornik: Mathematics, 2010, 201:7, 935–946

Bibliographic databases:

UDC: 517.938
MSC: Primary 37D05; Secondary 37B10, 37C50
Received: 02.11.2009

Citation: D. V. Anosov, “Extension of zero-dimensional hyperbolic sets to locally maximal ones”, Mat. Sb., 201:7 (2010), 3–14; Sb. Math., 201:7 (2010), 935–946

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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. Anosov D.V., “Intrinsic character of one property of hyperbolic sets”, J. Dyn. Control Syst., 16:4 (2010), 485–493  crossref  mathscinet  zmath  isi  scopus  scopus
    2. D. V. Anosov, “Local maximality of hyperbolic sets”, Proc. Steklov Inst. Math., 273 (2011), 23–24  mathnet  crossref  mathscinet  zmath  isi  elib
    3. Gorodetski A., “On stochastic sea of the standard map”, Comm. Math. Phys., 309:1 (2012), 155–192  crossref  mathscinet  zmath  adsnasa  isi  elib  scopus  scopus
    4. D. V. Anosov, “On trajectories entirely situated near a hyperbolic set”, Journal of Mathematical Sciences, 201:5 (2014), 553–565  mathnet  crossref  mathscinet
    5. Matheus C., Moreira C.G., Pujals E.R., “Axiom a Versus Newhouse Phenomena for Benedicks-Carleson Toy Models”, Ann. Sci. Ec. Norm. Super., 46:6 (2013), 857–878  crossref  mathscinet  zmath  isi  elib  scopus
    6. S. M. Aseev, V. M. Buchstaber, R. I. Grigorchuk, V. Z. Grines, B. M. Gurevich, A. A. Davydov, A. Yu. Zhirov, E. V. Zhuzhoma, M. I. Zelikin, A. B. Katok, A. V. Klimenko, V. V. Kozlov, V. P. Leksin, M. I. Monastyrskii, A. I. Neishtadt, S. P. Novikov, E. A. Sataev, Ya. G. Sinai, A. M. Stepin, “Dmitrii Viktorovich Anosov (obituary)”, Russian Math. Surveys, 70:2 (2015), 369–381  mathnet  crossref  crossref  mathscinet  adsnasa  isi  elib
    7. Da Luz A., “Hyperbolic Sets That Are Not Contained in a Locally Maximal One”, Discret. Contin. Dyn. Syst., 37:9 (2017), 4923–4941  crossref  mathscinet  zmath  isi  scopus
    8. Fisher T., Petty T., Tikhomirov S., “Nonlocally Maximal and Premaximal Hyperbolic Sets”, Modern Theory of Dynamical Systems: a Tribute to Dmitry Victorovich Anosov, Contemporary Mathematics, 692, eds. Katok A., Pesin Y., Hertz F., Amer Mathematical Soc, 2017, 83–99  crossref  mathscinet  zmath  isi  scopus
  • Математический сборник Sbornik: Mathematics (from 1967)
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