Geodesic
On the problem of topological classification of strange attractors of dynamical systems
Trudy Matematicheskogo Instituta imeni V.A. Steklova, Tome 57 (2002) no. 6, pp. 1163-1205 Cet article a éte moissonné depuis la source Math-Net.Ru

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This paper consists of two parts. The first, which is devoted to presenting results of Barge and Watkins, connects the closure of the union of the unstable manifolds of certain “Smale horseshoes” with Knaster continua and projections on them of Vietoris–van Dantzig solenoids. In the second part the homeomorphism problem for expanding attractors of codimension 1 is solved when the dimension of the manifold generating the dynamical system is greater than two. 
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R. V. Plykin. On the problem of topological classification of strange attractors of dynamical systems. Trudy Matematicheskogo Instituta imeni V.A. Steklova, Tome 57 (2002) no. 6, pp. 1163-1205. http://geodesic.mathdoc.fr/item/RM_2002_57_6_a2/

[1] J. Aarts, R. J. Fokkink, “The classification of solenoids”, Proc. Amer. Math. Soc., 111:4 (1991), 1161–1163 | DOI | MR | Zbl

[2] V. S. Afraimovich, V. V. Bykov, L. P. Shilnikov, “O prityagivayuschikh negrubykh predelnykh mnozhestvakh tipa attraktora Lorentsa”, Trudy MMO, 44 (1982), 150–212 | MR | Zbl

[3] P. S. Aleksandrov, Vvedenie v teoriyu mnozhestv i obschuyu topologiyu, Nauka, M., 1977 | MR

[4] D. Z. Arov, “O topologicheskom podobii avtomorfizmov i sdvigov kompaktnykh kommutativnykh grupp”, UMN, 18:5 (1963), 133–138 | MR | Zbl

[5] M. Barge, “Horseshoe maps and inverse limits”, Pacific J. Math., 121:1 (1986), 29–39 | MR | Zbl

[6] H. G. Bothe, Transversely Wild Attractors, Report R-MATH-03/83, Akad. Wiss. DDR, Inst. Math., Berlin, 1983 | MR | Zbl

[7] N. Burbaki, Obschaya topologiya. Chisla i svyazannye s nimi gruppy i prostranstva, Fizmatgiz, M., 1959 | MR

[8] D. Kassele, Vvedenie v teoriyu diofantovykh priblizhenii, IL, M., 1961

[9] R. J. Fokkink, The Structure of Trajectories, Amsterdam, 1991

[10] R. J. Franks, “Anosov diffeomorphisms”, Proc. Sympos. Pure Math., 14 (1970), 61–93 | MR | Zbl

[11] A. S. Gorodetski, Yu. S. Ilyashenko, “Minimal and strange attractors”, J. Bifurcations and Chaos, 6:6 (1996), 1177–1183 | DOI | MR

[12] J. Guckenheimer, R. F. Williams, “Structural stability of Lorenz attractors”, Inst. Hautes Étud. Sci. Publ. Math., 50 (1979), 59–72 | DOI | MR | Zbl

[13] E. Khyuitt, K. Ross, Abstraktnyi garmonicheskii analiz, 1, Nauka, M., 1975

[14] N. Klinshpont, “Neustoichivost yacheistoi struktury attraktorov Lorentsa”, Sbornik trudov uchaschikhsya Obninska, OIATE, Obninsk, 1989, 11–30

[15] N. Klinshpont, “Topologicheskii invariant attraktora Lorentsa”, UMN, 47:2 (1992), 195–196 | MR | Zbl

[16] E. N. Lorents, “Determinirovannoe neperiodicheskoe dvizhenie”, Strannye attraktory, Mir, M., 1981, 88–116

[17] M. C. McCord, “Inverse limit sequences with covering maps”, Trans. Amer. Math. Soc., 114 (1965), 197–209 | DOI | MR | Zbl

[18] J. Milnor, “On the concept of attractor”, Comm. Math. Phys., 99 (1985), 177–195 | DOI | MR | Zbl

[19] S. P. Novikov, “Topologiya sloenii”, Trudy MMO, 14 (1965), 248–278 | MR | Zbl

[20] A. Perov, I. Egle, “K teorii Puankare–Danzhua mnogomernykh differentsialnykh uravnenii”, Differents. uravneniya, 8:5 (1972), 801–810 | MR | Zbl

[21] Ya. B. Pesin, “Dynamical systems with generalized hyperbolic attractors: Hyperbolic, ergodic and topological properties”, Ergodic Theory Dynam. Systems, 12 (1992), 123–151 | DOI | MR | Zbl

[22] R. V. Plykin, “Istochniki i stoki $A$-diffeomorfizmov poverkhnostei”, Matem. sb., 94(136) (1974), 243–264 | MR | Zbl

[23] R. V. Plykin, “O suschestvovanii prityagivayuschikh (ottalkivayuschikh) tochek $A$-diffeomorfizmov proektivnoi ploskosti i butylki Kleina”, UMN, 32:3 (1977), 179 | MR | Zbl

[24] R. V. Plykin, “O giperbolicheskikh attraktorakh diffeomorfizmov”, UMN, 35:3 (1980), 94–104 | MR | Zbl

[25] R. V. Plykin, “O geometrii giperbolicheskikh attraktorov gladkikh kaskadov”, UMN, 39:6 (1984), 75–113 | MR | Zbl

[26] R. V. Plykin, A. Yu. Zhirov, “Some problems of attractors of dynamical systems”, Topology Appl., 54:1–3 (1993), 19–46 | DOI | MR | Zbl

[27] R. V. Plykin, “O strukture tsentralizatorov anosovskikh diffeomorfizmov tora”, UMN, 53:6 (1998), 259–260 | MR | Zbl

[28] R. V. Plykin, E. A. Sataev, S. V. Shlyachkov, “Strannye attraktory”, Itogi nauki i tekhniki. Sovr. problemy matem. Fundam. napr., 66, VINITI, M., 1991, 100–147 | MR | Zbl

[29] D. Rend, “Topologicheskaya klassifikatsiya attraktorov Lorentsa”, Strannye attraktory, Mir, M., 1981, 239–251

[30] D. Ruelle, F. Takens, “On the nature of turbulence”, Comm. Math. Phys., 20 (1971), 167–192 | DOI | MR | Zbl

[31] E. A. Sataev, “Invariantnye mery dlya giperbolicheskikh otobrazhenii s osobennostyami”, UMN, 47:1 (1992), 147–202 | MR | Zbl

[32] S. Smale, “Differentiable dynamical systems. I–IV”, Bull. Amer. Math. Soc., 73 (1967), 747–817 | DOI | MR

[33] S. Smale, “Mathematical problems for the next centure”, Math. Intelligencer, 20:2 (1997), 1–20 | MR

[34] Yu. Ustinov, “Algebraicheskie invarianty topologicheskoi sopryazhennosti solenoidov”, Matem. zametki, 42:1 (1987), 132–144 | MR | Zbl

[35] D. van Dantzig, “Über topologisch Homogene Kontinua”, Fund. Math., 15 (1930), 102–125

[36] L. Vietoris, “Über den höheren Zusammenhahg kompakter Räume und eine Klasse von zusammenhangstreuen Abbildungen”, Math. Ann., 97 (1927), 454–472 | DOI | MR | Zbl

[37] W. T. Watkins, “Homeomorphic classification of certain inverse limit spaces with open bonding maps”, Pacific J. Math., 103:2 (1982), 589–601 | MR | Zbl

[38] R. F. Williams, “The zeta function of an attractor”, Conference on the Topology of Manifolds, eds. J. Hocking, Prindle, Weber Schmidt, Boston, 1968, 155–161 | MR

[39] R. F. Williams, “Expanding attractors”, Publ. Inst. Hautes Étud. Sci. Publ. Math., 43 (1973), 169–203 | DOI | MR | Zbl

[40] R. F. Williams, O. Lanford, “The structure of Lorenz attractors”, Lecture Notes in Math., 615, 1977, 94–116 | MR | Zbl

[41] A. Yu. Zhirov, “Giperbolicheskie attraktory diffeomorfizmov orientiruemykh poverkhnostei, chast 1. Kodirovanie, klassifikatsiya i nakrytiya”, Matem. sb., 185:6 (1994), 3–50 | MR | Zbl

[42] A. Yu. Zhirov, “Giperbolicheskie attraktory diffeomorfizmov orientiruemykh poverkhnostei, chast 2. Perechisleniya i primeneniya k psevdoanosovskim diffeomorfizmam”, Matem. sb., 185:9 (1994), 29–80 | MR | Zbl

[43] A. Yu. Zhirov, “Giperbolicheskie attraktory diffeomorfizmov orientiruemykh poverkhnostei, chast 3. Algoritm klassifikatsii”, Matem. sb., 186:2 (1995), 69–82 | MR | Zbl

[44] A. Yu. Zhirov, “Complete combinatorial invariants of conjugacy of hyperbolic attractors of diffeomorphisms of surfaces”, J. Dynam. Control Systems, 6 (2000), 397–430 | DOI | MR | Zbl

[45] E. V. Zhuzhoma, “Singulyarnye sloeniya Riba na $n$-tore”, Matem. zametki, 30:1 (1981), 123–127 | MR