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Uspekhi Mat. Nauk, 1984, Volume 39, Issue 6(240), Pages 75–113 (Mi umn2504)  

This article is cited in 37 scientific papers (total in 39 papers)

On the geometry of hyperbolic attractors of smooth cascades

R. V. Plykin


Full text: PDF file (2847 kB)
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English version:
Russian Mathematical Surveys, 1984, 39:6, 85–131

Bibliographic databases:

UDC: 513.83+515.16
MSC: 37D40, 37C70, 37C55, 37C05, 53A07
Received: 25.03.1984

Citation: R. V. Plykin, “On the geometry of hyperbolic attractors of smooth cascades”, Uspekhi Mat. Nauk, 39:6(240) (1984), 75–113; Russian Math. Surveys, 39:6 (1984), 85–131

Citation in format AMSBIB
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\paper On~the~geometry of~hyperbolic attractors of~smooth cascades
\jour Uspekhi Mat. Nauk
\yr 1984
\vol 39
\issue 6(240)
\pages 75--113
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\transl
\jour Russian Math. Surveys
\yr 1984
\vol 39
\issue 6
\pages 85--131
\crossref{https://doi.org/10.1070/RM1984v039n06ABEH003182}
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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. S. Kh. Aranson, V. Z. Grines, “The topological classification of cascades on closed two-dimensional manifolds”, Russian Math. Surveys, 45:1 (1990), 1–35  mathnet  crossref  mathscinet  zmath  adsnasa  isi
    2. E. A. Sataev, “Invariant measures for hyperbolic maps with singularities”, Russian Math. Surveys, 47:1 (1992), 191–251  mathnet  crossref  mathscinet  zmath  adsnasa  isi
    3. Hisao Kato, “Expansive homeomorphisms in continuum theory”, Topology and its Applications, 45:3 (1992), 223  crossref
    4. Hisao Kato, “Embeddability into the plane and movability on inverse limits of graphs whose shift maps are expansive”, Topology and its Applications, 43:2 (1992), 141  crossref
    5. Leonardo Mora, Marcelo Viana, “Abundance of strange attractors”, Acta Math, 171:1 (1993), 1  crossref  mathscinet  zmath  isi
    6. Hisao Kato, “Concerning continuum-wise fully expansive homeomorphisms of continua”, Topology and its Applications, 53:3 (1993), 239  crossref
    7. A. Yu. Zhirov, “Hyperbolic attractors of diffeomorphisms of orientable surfaces”, Russian Acad. Sci. Sb. Math., 82:1 (1995), 135–174  mathnet  crossref  mathscinet  zmath  isi
    8. S. Kh. Aranson, E. V. Zhuzhoma, “On the structure of quasiminimal sets of foliations on surfaces”, Russian Acad. Sci. Sb. Math., 82:2 (1995), 397–424  mathnet  crossref  mathscinet  zmath  isi
    9. A. Yu. Zhirov, “Hyperbolic attractors of diffeomorphisms of orientable surfaces”, Russian Acad. Sci. Sb. Math., 83:1 (1995), 23–65  mathnet  crossref  mathscinet  zmath  isi
    10. A. Yu. Zhirov, R. V. Plykin, “On the relationship between one-dimensional hyperbolic attractors of surface diffeomorphisms and generalized pseudo-Anosov diffeomorphisms”, Math. Notes, 58:1 (1995), 779–781  mathnet  crossref  mathscinet  zmath  isi
    11. Vincent Naudot, “Strange attractor in the unfolding of an inclination-flip homoclinic orbit”, Ergod Th Dynam Sys, 16:5 (1996), 1071  crossref
    12. V. Z. Grines, “On the topological classification of structurally stable diffeomorphisms of surfaces with one-dimensional attractors and repellers”, Sb. Math., 188:4 (1997), 537–569  mathnet  crossref  crossref  mathscinet  zmath  isi
    13. A. Yu. Zhirov, “Solenoidal representations and the homology of hyperbolic attractors of diffeomorphisms of surfaces”, Sb. Math., 188:6 (1997), 799–821  mathnet  crossref  crossref  mathscinet  zmath  isi  elib
    14. V. Z. Grines, “A representation of one-dimensional attractors of $A$-diffeomorphisms by hyperbolic homeomorphisms”, Math. Notes, 62:1 (1997), 64–73  mathnet  crossref  crossref  mathscinet  zmath  isi  elib
    15. Hisao Kato, “Expansive homeomorphisms on surfaces with holes”, Topology and its Applications, 82:1-3 (1998), 267  crossref
    16. A. Yu. Zhirov, “Examples of one-dimensional hyperbolic attractors on nonorientable surfaces”, Math. Notes, 65:3 (1999), 390–393  mathnet  crossref  crossref  mathscinet  zmath  isi
    17. R. V. Plykin, “On the problem of topological classification of strange attractors of dynamical systems”, Russian Math. Surveys, 57:6 (2002), 1163–1205  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib
    18. V. Z. Grines, E. V. Zhuzhoma, “Structurally stable diffeomorphisms with basis sets of codimension one”, Izv. Math., 66:2 (2002), 223–284  mathnet  crossref  crossref  mathscinet  zmath
    19. D. V. Anosov, E. V. Zhuzhoma, “Asymptotic Behavior of Covering Curves on the Universal Coverings of Surfaces”, Proc. Steklov Inst. Math., 238 (2002), 1–46  mathnet  mathscinet  zmath
    20. E. V. Zhuzhoma, V. S. Medvedev, “On non-orientable two-dimensional basic sets on 3-manifolds”, Sb. Math., 193:6 (2002), 869–888  mathnet  crossref  crossref  mathscinet  zmath  isi
    21. Christopher Mouron, “Expansive homeomorphisms and indecomposable subcontinua”, Topology and its Applications, 126:1-2 (2002), 13  crossref
    22. R. V. Plykin, “Strange attractors of $A$-cascades”, Russian Math. Surveys, 59:2 (2004), 390–392  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi
    23. A. Yu. Zhirov, “Combinatorics of One-Dimensional Hyperbolic Attractors of Diffeomorphisms of Surfaces”, Proc. Steklov Inst. Math., 244 (2004), 132–200  mathnet  mathscinet  zmath
    24. D. V. Anosov, E. V. Zhuzhoma, “Nonlocal asymptotic behavior of curves and leaves of laminations on universal coverings”, Proc. Steklov Inst. Math., 249 (2005), 1–221  mathnet  mathscinet  zmath
    25. Grines, VZ, “Expanding attractors”, Regular & Chaotic Dynamics, 11:2 (2006), 225  crossref  mathscinet  zmath  adsnasa  isi  elib
    26. Christopher Mouron, “Expansive homeomorphisms and plane separating continua”, Topology and its Applications, 155:9 (2008), 1000  crossref
    27. E. V. Zhuzhoma, V. S. Medvedev, “Surface Basic Sets with Wildly Embedded Supporting Surfaces”, Math. Notes, 85:3 (2009), 353–365  mathnet  crossref  crossref  mathscinet  zmath  isi  elib
    28. E. V. Zhuzhoma, N. V. Isaenkova, “Classification of One-Dimensional Expanding Attractors”, Math. Notes, 86:3 (2009), 333–341  mathnet  crossref  crossref  mathscinet  zmath  isi  elib
    29. AARON W. BROWN, “Constraints on dynamics preserving certain hyperbolic sets”, Ergod Th Dynam Sys, 2010, 1  crossref  isi
    30. A. Yu. Loskutov, “Fascination of chaos”, Phys. Usp., 53:12 (2010), 1257–1280  mathnet  crossref  crossref  isi  elib
    31. Plykin R.V., Klinshpont N.E., “Strange attractors. Topologic, geometric and algebraic aspects”, Regular & Chaotic Dynamics, 15:2–3 (2010), 335–347  crossref  isi  elib
    32. D. V. Anosov, M. Ya. Antonovskii, V. M. Buchstaber, V. Z. Grines, A. Yu. Zhirov, E. V. Zhuzhoma, N. È. Klinshpont, A. A. Mal'tsev, E. A. Sataev, Ya. G. Sinai, A. M. Stepin, L. P. Shil'nikov, “Romen Vasil'evich Plykin (obituary)”, Russian Math. Surveys, 66:3 (2011), 627–631  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib  elib
    33. A. Yu. Loskutov, A. V. Popkova, “Stabilization of chaotic oscillations in systems with a hyperbolic-type attractor”, JETP Letters, 94:1 (2011), 86–90  mathnet  crossref  isi
    34. Hisao Kato, “On indecomposability and composants of chaotic continua, II”, Topology and its Applications, 158:5 (2011), 647  crossref
    35. A. Yu. Zhirov, “How Many Different Cascades on a Surface Can Have Coinciding Hyperbolic Attractors?”, Math. Notes, 94:1 (2013), 96–106  mathnet  crossref  crossref  mathscinet  zmath  isi  elib  elib
    36. A. G. Fedotov, “On the Realization of the Generalized Solenoid as a Hyperbolic Attractor of Sphere Diffeomorphisms”, Math. Notes, 94:5 (2013), 681–691  mathnet  crossref  crossref  mathscinet  zmath  isi  elib  elib
    37. Dmitry Turaev, “Maps Close to Identity and Universal Maps in the Newhouse Domain”, Commun. Math. Phys, 2015  crossref
    38. 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
    39. A. G. Fedotov, “On the Solenoidal Representation of the Hyperbolic Attractor of a Diffeomorphism of the Sphere”, Math. Notes, 101:1 (2017), 181–183  mathnet  crossref  crossref  mathscinet  isi  elib
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