On the problem of topological classification of strange attractors of dynamical systems

Russian Mathematical Surveys

Volumn 57, Issue 6, 2002, Pages 1163-1205

  Plykin, R V ^a  

^a NONE

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EID: 0038326678     PISSN: 00360279     EISSN: 14684829     Source Type: Journal    
DOI: 10.1070/RM2002v057n06ABEH000574     Document Type: Review

References (53)

1. J. M. Aarts and R. J. Fokkink, "The classification of solenoids", Proc. Amer. Math. Soc. 111 (1991), 1161-1163.
2. V. S. Afraimovich, V. V. Bykov, and L. P. Shil'nikov, "On attracting structurally unstable limit sets of Lorenz attractor type", Trudy Moskov. Mat. Obshch. 44 (1982), 150-212; English transl., Trans. Moscow Math. Soc. 1988, no. 2, 153-216.
3. P. S. Aleksandrov, Introduction to set theory and general topology, Nauka, Moscow 1977; German transl., VEB Deutscher Verlag der Wissenschaften, Berlin 1984.
4. D. Z. Arov, "Topological similarity of automorphisms and translations of compact commutative groups", Uspekhi Mat. Nauk 18:5 (1963), 133-138. (Russian)
5. M. Barge, "Horseshoe maps and inverse limits", Pacific J. Math. 121 (1986), 29-39.
6. H. G. Bothe, "Transversely wild attractors", Report R-MATH-03/83, Akad. Wiss. DDR, Inst. Math., Berlin 1983.
7. N. Bourbaki, Topologie générale, 2nd ed., Actualités Sci. Ind., nos. 916, 1029, Hermann, Paris 1942, 1947; Russian transl. of Chaps. IV, VI, VII, Gos. Izdat. Fiz. Mat. Lit., Moscow 1959.
8. N. Bourbaki, Topologie générale, 2nd ed., Actualités Sci. Ind., nos. 916, 1029, Hermann, Paris 1942, 1947; Russian transl. of Chaps. IV, VI, VII, Gos. Izdat. Fiz. Mat. Lit., Moscow 1959.
9. J. W. S. Cassels, Introduction to the theory of Diophantine approximation, Cambridge Univ. Press, Cambridge 1957; Russian transl., Inostr. Lit., Moscow 1961.
10. J. W. S. Cassels, Introduction to the theory of Diophantine approximation, Cambridge Univ. Press, Cambridge 1957; Russian transl., Inostr. Lit., Moscow 1961.
11. R. J. Fokkink, The structure of trajectories, PhD thesis, Technical University of Delft 1991.
12. R. J. Franks, "Anosov diffeomorphisms", Proc. Sympos. Pure Math. 14 (1970), 61-93.
13. A. S. Gorodetski and Yu. S. Ilyashenko, "Minimal and strange attractors", Internat. J. Bifur. Chaos Appl. Sci. Engrg. 6 (1996), 1177-1183.
14. J. Guckenheimer and R. F. Williams, "Structural stability of Lorenz attractors", Inst. Hautes Étud. Sci. Publ. Math. 50 (1979), 59-72.
15. E. Hewitt and K. A. Ross, Abstract harmonic analysis, vol. I: Structure of topological groups. Integration theory, group representations., Academic Press, New York 1963; Russian transl., Nauka, Moscow 1975.
16. N. Klinshpont, "The instability of the cellular structure of Lorenz attractors", Collection of papers by Obninsk students, Obninsk Atomic Energy Institute, Obninsk 1989, pp. 11-30. (Russian)
17. N. Klinshpont, "A topological invariant of the Lorenz attractor", Uspekhi Mat. Nauk 47:2 (1992), 195-196; English transl., Russian Math. Surveys 47:2 (1992), 221-223.
18. E. M. Lorenz, "Deterministic nonperiodic flow", J. Atmospheric Sci. 20 (1963), 130-141; Russian transl., Strange attractors, Mir, Moscow 1981, pp. 88-116.
19. M. C. McCord, "Inverse limit sequences with covering maps", Trans. Amer. Math. Soc. 114 (1965), 197-209.
20. J. Milnor, "On the concept of attractor", Comm. Math. Phys. 99 (1985), 177-195.
21. S. P. Novikov, "The topology of foliations", Trudy Moskov. Mat. Obshch. 14 (1965), 248-278; English transl., Trans. Moscow Math. Soc. 14 (1965), 268-304.
22. A. I. Perov and I. Yu. Egle, "On the Poincaré-Denjoy theory of multidimensional differential equations", Differentsial'nye Uravneniya 8 (1972), 801-810; English transl., Differential Equations 8 (1972), 608-615.
23. Ya. B. Pesin, "Dynamical systems with generalized hyperbolic attractors: Hyperbolic, ergodic and topological properties", Ergodic Theory Dynam. Systems 12 (1992), 123-151.
24. R. V. Plykin, "Sources and sinks of A-diffeomorphisms of surfaces", Mat. Sb. 94 (1974), 243-264; English transl., Math. USSR-Sb. 23 (1974), 233-253.
25. R. V. Plykin, "Sources and sinks of A-diffeomorphisms of surfaces", Mat. Sb. 94 (1974), 243-264; English transl., Math. USSR-Sb. 23 (1974), 233-253.
26. R. V. Plykin, "The existence of attracting (repelling) periodic points of axiom A diffeomorphisms of the projective plane and of the Klein bottle", Uspekhi Mat. Nauk 32:3 (1977), 179. (Russian)
27. R. V. Plykin, "Hyperbolic attractors of diffeomorphisms", Internat. Topology Conf. (Moscow State Univ., Moscow 1979), Uspekhi Mat. Nauk 35:3 (1980), 94-104; English transl., Russian Math. Surveys 35:3 (1980), 109-121.
28. R. V. Plykin, "Hyperbolic attractors of diffeomorphisms", Internat. Topology Conf. (Moscow State Univ., Moscow 1979), Uspekhi Mat. Nauk 35:3 (1980), 94-104; English transl., Russian Math. Surveys 35:3 (1980), 109-121.
29. R. V. Plykin, "On the geometry of hyperbolic attractors of smooth cascades", Uspekhi Mat. Nauk 39:6 (1984), 75-113; English transl., Russian Math. Surveys 39:6 (1984), 85-131.
30. R. V. Plykin and A. Yu. Zhirov, "Some problems of attractors of dynamical systems", Topology Appl. 54 (1993), 19-46.
31. R. V. Plykin, "On the structure of centralizers of Anosov diffeomorphisms of a torus", Uspekhi Mat. Nauk 53:6 (1998), 259-260; English transl., Russian Math. Surveys 53 (1998), 1363-1364.
32. R. V. Plykin, E. A. Sataev, and S. V. Shlyachkov, "Strange attractors", Itogi Nauki i Tekhniki: Sovremennye Problemy Mat.: Fundamental'nye Napravleniya, vol. 66, VINITI, Moscow 1991, pp. 100-147; English transl., Dynamical systems IX, Encyclopaedia Math. Sci., vol. 66, Springer-Verlag, Berlin, 1995, pp. 93-139.
33. R. V. Plykin, E. A. Sataev, and S. V. Shlyachkov, "Strange attractors", Itogi Nauki i Tekhniki: Sovremennye Problemy Mat.: Fundamental'nye Napravleniya, vol. 66, VINITI, Moscow 1991, pp. 100-147; English transl., Dynamical systems IX, Encyclopaedia Math. Sci., vol. 66, Springer-Verlag, Berlin, 1995, pp. 93-139.
34. D. Rand, "The topological classification of Lorenz attractors", Proc. Cambridge Philos. Soc. 83 (1978), 451-460; Russian transl., Strange Attractors, Mir, Moscow 1981, pp. 239-251.
35. D. Rand, "The topological classification of Lorenz attractors", Proc. Cambridge Philos. Soc. 83 (1978), 451-460; Russian transl., Strange Attractors, Mir, Moscow 1981, pp. 239-251.
36. D. Ruelle and F. Takens, "On the nature of turbulence", Comm. Math. Phys. 20 (1971), 167-192.
37. E. A. Sataev, "Invariant measures for hyperbolic mappings with singularities", Uspekhi Mat. Nauk 47:1 (1992), 147-202; English transl., Russian Math. Surveys 47:1 (1992), 191-251.
38. S. Smale, "Differentiable dynamical systems. I-IV", Bull. Amer. Math. Soc. 73 (1967), 747-817.
39. S. Smale, "Mathematical problems for the next century", Math. Intelligencer 20:2 (1997), 1-20.
40. Yu. Ustinov, "Algebraic invariants of the topological conjugacy classes of solenoids", Mat. Zametki 42:1 (1987), 132-144; English transl., Math. Notes 42 (1987), 583-590.
41. D. van Dantzig, "Über topologisch homogene Kontinua", Fund. Math. 15 (1930), 102-125.
42. L. Vietoris, "Über den höheren Zusammenhang kompakter Räume und eine Klasse von zusammenhangstreuen Abbildungen", Math. Ann. 97 (1927), 454-472.
43. W. T. Watkins, "Homeomorphic classification of certain inverse limit spaces with open bonding maps", Pacific J. Math. 103 (1982), 589-601.
44. R. F. Williams, "The zeta function of an attractor", Proc. Conf. on the Topology of Manifolds (J. Hocking, ed.), Prindle, Weber, and Schmidt, Boston 1968, pp. 155-161.
45. R. F. Williams, "Expanding attractors", Publ. Inst. Hautes Étud. Sci. Publ. Math. 43 (1973), 169-203.
46. R. F. Williams and O. Lanford, "The structure of Lorenz attractors", Lecture Notes in Math. 615 (1977), 94-116.
47. A. Yu. Zhirov, "Hyperbolic attractors of diffeomorphisms of oriented surfaces. 1. Coding, classification, and coverings", Mat. Sb. 185:6 (1994), 3-50; English transl., Russian Acad. Sci. Sb. Math. 82 (1995), 135-174.
48. A. Yu. Zhirov, "Hyperbolic attractors of diffeomorphisms of oriented surfaces. 1. Coding, classification, and coverings", Mat. Sb. 185:6 (1994), 3-50; English transl., Russian Acad. Sci. Sb. Math. 82 (1995), 135-174.
49. A. Yu. Zhirov, "Hyperbolic attractors of diffeomorphisms of oriented surfaces. II. Enumeration and application to pseudo-Anosov diffeomorphisms", Mai. Sb. 185:9 (1994), 29-80; English transl., Russian Acad. Sci. Sb. Math. 83 (1996), 23-65.
50. A. Yu. Zhirov, "Hyperbolic attractors of diffeomorphisms of oriented surfaces. II. Enumeration and application to pseudo-Anosov diffeomorphisms", Mai. Sb. 185:9 (1994), 29-80; English transl., Russian Acad. Sci. Sb. Math. 83 (1996), 23-65.
51. A. Yu. Zhirov, "Hyperbolic attractors of diffeomorphisms of oriented surfaces. III. A classification algorithm", Mat. Sb. 186:2 (1995), 69-82; English transl., Sb. Math. 186 (1995), 221-244.
52. A. Yu. Zhirov, "Complete combinatorial invariants of conjugacy of hyperbolic attractors of diffeomorphisms of surfaces", J. Dynam. Control Systems 6 (2000), 397-430.
53. E. V. Zhuzhoma, "Singular Reeb foliations on an n-dimensional torus", Mat. Zametki 30:1 (1981), 123-128; English transl., Math. Notes 30 (1982), 549-551.