Diffeomorphisms of 2-manifolds with one-dimensional spaciously situated basic sets

V. Z. Grines ; E. D. Kurenkov

Izvestiya. Mathematics , Tome 84 (2020) no. 5, pp. 862-909.

Résumé

We consider orientation-preserving $A$-diffeomorphisms of orientable surfaces of genus greater than one with a one-dimensional spaciously situated perfect attractor. We show that the topological classification of restrictions of diffeomorphisms to such basic sets can be reduced to that of pseudo-Anosov homeomorphisms with a distinguished set of saddles. In particular, we prove a result announced by Zhirov and Plykin, which gives a topological classification of the $A$-diffeomorphisms of the surfaces under discussion under the additional assumption that the non-wandering set consists of a one-dimensional spaciously situated attractor and zero-dimensional sources.

Détail

Keywords: one-dimensional basic set, perfect attractor, spaciously situated set.
Mots-clés : axiom $A$

Bibliographie

[1] S. Smale, “Differentiable dynamical systems”, Bull. Amer. Math. Soc., 73:6 (1967), 747–817 | DOI | MR | MR | Zbl

[2] V. Z. Grines, E. D. Kurenkov, “Classification of one-dimensional attractors of diffeomorphisms of surfaces by means of pseudo-Anosov homeomorphisms”, Dokl. Math., 99:2 (2019), 137–139 | DOI | DOI | Zbl

[3] R. V. Plykin, “Sources and sinks of $A$-diffeomorphisms of surfaces”, Math. USSR-Sb., 23:2 (1974), 233–253 | DOI | MR | Zbl

[4] 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 | DOI | MR | Zbl

[5] C. Bonatti, R. Langevin, Difféomorphismes de Smale des surfaces, Astérisque, 250, Soc. Math. France, Paris, 1998, viii+235 pp. | MR | Zbl

[6] A. A. G. Ruas, Atratores hiperbolicos de codimensao um e classes de isotopia em superficies, Tese de Doutoramento, IMPA, Rio de Janeiro, 1982

[7] M. Barge, B. F. Martensen, “Classification of expansive attractors on surfaces”, Ergodic Theory Dynam. Systems, 31:6 (2011), 1619–1639 | DOI | MR | Zbl

[8] V. Z. Grines, “On the topological conjugacy of diffeomorphisms of a two-dimensional manifold on one-dimensional orientable basic sets. II”, Trans. Moscow Math. Soc., 34 (1978), 237–245 | MR | Zbl

[9] V. Z. Grines, R. V. Plykin, “Topological classification of amply situated attractors of $A$-diffeomorphisms of surfaces”, Methods of qualitative theory of differential equations and related topics, Amer. Math. Soc. Transl. Ser. 2, 200, Adv. Math. Sci., 48, Amer. Math. Soc., Providence, RI, 2000, 135–148 | DOI | MR | Zbl

[10] A. Yu. Zhirov, Topologicheskaya sopryazhennost psevdoanosovskikh gomeomorfizmov, MTsNMO, M., 2013, 368 pp.

[11] V. Z. Grines, “On topological conjugacy of diffeomorphisms of a two-dimensional manifold onto one-dimensional orientable basic sets. I”, Trans. Moscow Math. Soc., 32 (1977), 31–56 | MR | Zbl

[12] R. V. Plykin, “On the geometry of hyperbolic attractors of smooth cascades”, Russian Math. Surveys, 39:6 (1984), 85–131 | DOI | MR | Zbl

[13] V. Z. Grinec, Kh. Kh. Kalai, “O topologicheskoi ekvivalentnosti diffeomorfizmov s netrivialnymi bazisnymi mnozhestvami na dvumernykh mnogoobraziyakh”, Metody kachestvennoi teorii differentsialnykh uravnenii, Mezhvuz. temat. sb. nauch. tr., ed. E. A. Leontovich-Andronova, Iz-vo Gork. gos. un-t, Gorkii, 1988, 40–49

[14] S. Kh. Aranson, V. Z. Grines, “The topological classification of cascades on closed two-dimensional manifolds”, Russian Math. Surveys, 45:1 (1990), 1–35 | DOI | MR | Zbl

[15] V. Z. Grines, “Topological classification of one-dimensional attractors and repellers of $A$-diffeomorphisms of surfaces by means of automorphisms of fundamental groups of supports”, J. Math. Sci. (N.Y.), 95:5 (1999), 2523–2545 | DOI | MR | Zbl

[16] V. Z. Grines, “On topological classification of $A$-diffeomorphisms of surfaces”, J. Dynam. Control Systems, 6:1 (2000), 97–126 | DOI | MR | Zbl

[17] A. Yu. Zhirov, “Hyperbolic attractors of diffeomorphisms of orientable surfaces”, Russian Acad. Sci. Sb. Math., 82:1 (1995), 135–174 | DOI | DOI | MR | Zbl

[18] A. Yu. Zhirov, “Hyperbolic attractors of diffeomorphisms of orientable surfaces”, Russian Acad. Sci. Sb. Math., 83:1 (1995), 23–65 | DOI | MR | Zbl

[19] A. Yu. Zhirov, “Hyperbolic attractors of diffeomorphisms of orientable surfaces. Part 3. Classification algorithm”, Sb. Math., 186:2 (1995), 221–244 | DOI | MR | Zbl

[20] V. Z. Grines, Kh. Kh. Kalai, “Diffeomorphisms of two-dimensional manifolds with spatially situated basic sets”, Russian Math. Surveys, 40:1 (1985), 221–222 | DOI | MR | Zbl

[21] 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 | DOI | DOI | MR | Zbl

[22] J. Nielsen, “Untersuchungen zur Topologie der geschlossenen zweiseitigen Flächen”, Acta Math., 50:1 (1927), 189–358 | DOI | MR | Zbl

[23] S. H. Aranson, V. Z. Grines, “On some invariants of dynamical systems on two-dimensional manifolds (necessary and sufficient conditions for the topological equivalence of transitive dynamical systems)”, Math. USSR-Sb., 19:3 (1973), 365–393 | DOI | MR | Zbl

[24] R. V. Plykin, “The topology of basis sets for Smale diffeomorphisms”, Math. USSR-Sb., 13:2 (1971), 297–307 | DOI | MR | Zbl

[25] A. J. Casson, S. A. Bleiler, Automorphisms of surfaces after Nielsen and Thurston, London Math. Soc. Stud. Texts, 9, Cambridge Univ. Press, Cambridge, 1988, iv+105 pp. | DOI | MR | Zbl

[26] A. Fathi, F. Laudenbach, V. Poénaru, Thurston's work on surfaces, Transl. from the French, Math. Notes, 48, Princeton Univ. Press, Princeton, NJ, 2012, xvi+254 pp. | MR | Zbl

[27] R. Bowen, “Periodic points and measures for Axiom $A$ diffeomorphisms”, Trans. Amer. Math. Soc., 154 (1971), 377–397 | DOI | MR | Zbl

[28] D. V. Anosov, “Ob odnom klasse invariantnykh mnozhestv gladkikh dinamicheskikh sistem”, Tpudy V mezhdunapodnoi konfepentsii po nelineinym kolebaniyam, v. 2, Kachestvennye metody, In-t matem. AN USSR, Kiev, 1970, 39–45 | Zbl

[29] V. Z. Grines, O. V. Pochinka, Vvedenie v topologicheskuyu klassifikatsiyu kaskadov na mnogoobraziyakh razmernosti dva i tri, NITs “Regulyarnaya i khaoticheskaya dinamika”, In-t kompyuternykh issledovanii, M.–Izhevsk, 2011, 424 pp.

[30] D. B. A. Epstein, “Curves on 2-manifolds and isotopies”, Acta Math., 115:1 (1966), 83–107 | DOI | MR | Zbl

[31] S. H. Aranson, V. Z. Grines, “On the representation of minimal sets of currents on two-dimensional manifolds by geodesics”, Math. USSR-Izv., 12:1 (1978), 103–124 | DOI | MR | Zbl

[32] D. V. Anosov, S. Kh. Aranson, V. Z. Grines, R. V. Plykin, E. A. Sataev, A. V. Safonov, V. V. Solodov, A. N. Starkov, A. M. Stepin, S. V. Shlyachkov, Dynamical systems IX. Dynamical systems with hyperbolic behavior, Encyclopaedia Math. Sci., 66, Springer, Berlin, 1995, vi+235 pp. | DOI | MR | MR | Zbl

[33] 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 | MR | Zbl

[34] P. S. Alexandroff, Einführung in die Mengenlehre und in die allgemeine Topologie, Hochschulbücher für Math., 85, Deutscher Verlag Wissensch., Berlin, 1984, 336 pp. | MR | MR