Surface Basic Sets with Wildly Embedded Supporting Surfaces
Matematičeskie zametki, Tome 85 (2009) no. 3, pp. 356-372.

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The situation where a “nice” diffeomorphism $f$ of a 3-manifold has a wildly embedded invariant surface $M$ for which the restriction $g=f|_M\colon M\to M$ is “nice” is considered.
Keywords: Anosov diffeomorphism, 3-manifold, Morse–Smale diffeomorphism, nonwandering set, surface basic set, wildly embedded supporting surface, conjugate diffeomorphisms.
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E. V. Zhuzhoma; V. S. Medvedev. Surface Basic Sets with Wildly Embedded Supporting Surfaces. Matematičeskie zametki, Tome 85 (2009) no. 3, pp. 356-372. http://geodesic.mathdoc.fr/item/MZM_2009_85_3_a4/

[1] D. V. Anosov, “Gladkie dinamicheskie sistemy. Gl. 1. Iskhodnye ponyatiya”, Dinamicheskie sistemy – 1, Itogi nauki i tekhn. Ser. Sovrem. probl. mat. Fundam. napravleniya, 1, VINITI, M., 1985, 156–178

[2] 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, “Dinamicheskie sistemy s giperbolicheskim povedeniem”, Dinamicheskie sistemy – 9, Itogi nauki i tekhn. Ser. Sovrem. probl. mat. Fundam. napravleniya, 66, VINITI, M., 1991, 5–242 | MR | Zbl

[3] S. Smale, “Differentiable dynamical systems”, Bull. Amer. Math. Soc., 73:1 (1967), 747–817 ; С. Смейл, “Дифференцируемые динамические системы”, УМН, 25:1 (1970), 113–185 | DOI | MR | Zbl | MR

[4] R. V. Plykin, “O topologii bazisnykh mnozhestv diffeomorfizmov Smeila”, Matem. sb., 84:2 (1971), 301–312 | MR | Zbl

[5] R. F. Williams, “Expanding attractors”, Inst. Hautes Études Sci. Publ. Math., 43 (1974), 169–203 | DOI | MR | Zbl

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

[7] V. Z. Grines, Kh. Kh. Kalai, Topologicheskaya klassifikatsiya bazisnykh mnozhestv bez par sopryazhennykh tochek A-diffeomorfizmov na poverkhnostyakh, dep. v VINITI 1137-V88

[8] C. Bonatti, R. Langevin, Difféomorphismes de Smale des surfaces, With the collaboration of E. Jeandenans, Astérisque, 250, Soc. Math. France, Paris, 1998 | MR | Zbl

[9] R. Bowen, J. Franks, “Homology for zero-dimensional nonwandering sets”, Ann. of Math. (2), 106:1 (1977), 73–92 | DOI | MR | Zbl

[10] J. M. Franks, “Constructing structurally stable diffeomorphisms”, Ann. of Math. (2), 105:2 (1977), 343–359 | DOI | MR | Zbl

[11] J. M. Franks, Homology and Dynamical Systems, CBMS Regional Conf. Ser. in Math., 49, Amer. Math. Soc., Providence, RI, 1982 | MR | Zbl

[12] R. F. Williams, “Classification of subshifts of finite type”, Ann. of Math. (2), 98:1 (1973), 120–153 ; “Errata to "Classification of subshifts of finite type”, Ann. of Math. (2), 99:2 (1974), 380–381 | DOI | MR | Zbl | DOI

[13] V. Z. Grines, E. V. Zhuzhoma, “Expanding attractors”, Regul. Chaotic Dyn., 11:2 (2006), 225–246 | DOI | MR | Zbl

[14] V. Z. Grines, V. S. Medvedev, E. V. Zhuzhoma, “O poverkhnostnykh attraktorakh i repellerakh na 3-mnogoobraziyakh”, Matem. zametki, 78:6 (2005), 813–826 | MR | Zbl

[15] L. V. Keldysh, Topologicheskie vlozheniya v evklidovo prostranstvo, Tr. MIAN, 81, Nauka, M., 1966

[16] J. Kaplan, J. Mallet-Paret, J. Yorke, “The Lyapunov dimension of a nowhere differentiable attracting torus”, Ergodic Theory Dynam. Systems, 4:2 (1984), 261–281 | DOI | MR | Zbl

[17] D. Pixton, “Wild unstable manifolds”, Topology, 16:2 (1977), 167–172 | DOI | MR | Zbl

[18] R. Mañé, “A proof of the $C^1$ stability conjecture”, Inst. Hautes Études Sci. Publ. Math., 66 (1988), 161–210 | DOI | MR | Zbl

[19] C. Robinson, “Structural stability of $C^1$ diffeomorphisms”, J. Differential Equations, 22:1 (1976), 28–73 | DOI | MR | Zbl

[20] M. Hirsch, J. Palis, C. Pugh, M. Shub, “Neighborhoods of hyperbolic sets”, Invent. Math., 9:2 (1970), 121–134 | DOI | MR | Zbl

[21] J. Palis, “On the $C^1$ $\Omega$-stability conjecture”, Inst. Hautes Études Sci. Publ. Math., 66 (1988), 211–215 | DOI | MR | Zbl

[22] S. Smale, “Morse inequalities for a dynamical system”, Bull. Amer. Math. Soc., 66 (1960), 43–49 | DOI | MR | Zbl

[23] T. M. Cherry, “Analytic quasi-periodic curves of discontinuous type on a torus”, Proc. London Math. Soc. (2), 44:3 (1938), 175–215 | DOI | Zbl

[24] F. W. Wilson jr., “On the minimal sets of non-singular vector fields”, Ann. of Math. (2), 84:3 (1966), 529–536 | DOI | MR | Zbl

[25] G. Fleitas, “Classification of gradient-like flows in dimension two and three”, Bol. Soc. Brasil. Mat., 6:2 (1975), 155–183 | DOI | MR | Zbl

[26] S. Smale, “The generalized Poincaré conjecture in higher dimensions”, Bull. Amer. Math. Soc., 66 (1960), 373–375 | DOI | MR | Zbl

[27] S. Smale, “On gradient dynamical systems”, Ann. of Math. (2), 74:1 (1961), 199–206 | DOI | MR | Zbl

[28] S. Smale, “Generalized Poincaré's conjecture in dimensions greater than four”, Ann. of Math. (2), 74:2 (1961), 391–406 | DOI | MR | Zbl

[29] R. H. Fox, E. Artin, “Some wild cells and spheres in three-dimensional space”, Ann. of Math. (2), 49:4 (1948), 979–990 | DOI | MR | Zbl

[30] C. Bonatti, V. Grines, “Knots as topological invariants for gradient-like diffeomorphisms of the sphere $S^3$”, J. Dynam. Control Systems, 6:4 (2000), 579–602 | DOI | MR | Zbl

[31] Dzh. Milnor, Teoriya Morsa, Mir, M., 1965 | MR | Zbl