Mat. Sb.:
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
References: PDF file HTML file
Sbornik: Mathematics, 2010, 201:7, 935–946
UDC: 517.938
MSC: Primary 37D05;	Secondary 37B10, 37C50
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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This publication is cited in the following articles:
Anosov D.V., "Intrinsic character of one property of hyperbolic sets", J. Dyn. Control Syst., 16:4 (2010), 485–493
D. V. Anosov, "Local maximality of hyperbolic sets", Proc. Steklov Inst. Math., 273 (2011), 23–24
Gorodetski A., "On stochastic sea of the standard map", Comm. Math. Phys., 309:1 (2012), 155–192
D. V. Anosov, "On trajectories entirely situated near a hyperbolic set", Journal of Mathematical Sciences, 201:5 (2014), 553–565
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
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
Da Luz A., "Hyperbolic Sets That Are Not Contained in a Locally Maximal One", Discret. Contin. Dyn. Syst., 37:9 (2017), 4923–4941
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
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