Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 
 

 

Quasi-Anosov diffeomorphisms of 3-manifolds


Authors: T. Fisher and M. Rodriguez Hertz
Journal: Trans. Amer. Math. Soc. 361 (2009), 3707-3720
MSC (2000): Primary 37D05, 37D20
DOI: https://doi.org/10.1090/S0002-9947-09-04687-X
Published electronically: February 10, 2009
MathSciNet review: 2491896
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: In 1969, Hirsch posed the following problem: given a diffeomorphism $ f:N\to N$ and a compact invariant hyperbolic set $ \Lambda$ of $ f$, describe the topology of $ \Lambda$ and the dynamics of $ f$ restricted to $ \Lambda$. We solve the problem where $ \Lambda=M^3$ is a closed $ 3$-manifold: if $ M^3$ is orientable, then it is a connected sum of tori and handles; otherwise it is a connected sum of tori and handles quotiented by involutions.

The dynamics of the diffeomorphisms restricted to $ M^3$, called quasi-Anosov diffeomorphisms, is also classified: it is the connected sum of DA-diffeomorphisms, quotiented by commuting involutions.


References [Enhancements On Off] (What's this?)

  • 1. J. Franks, Anosov diffeomorphisms, Global Analysis Proc. Symp. Pure Math. 14 AMS Providence, Rhode Island (1970) 61-93 MR 0271990 (42:6871)
  • 2. J. Franks, Invariant sets of hyperbolic toral automorphisms, Amer. J. Math. 99 (1977) 1089-1095 MR 0482846 (58:2891)
  • 3. J. Franks, C. Robinson, A quasi-Anosov diffeomorphism that is not Anosov, Trans. AMS 223 (1976) 267-278 MR 0423420 (54:11399)
  • 4. V. Grines, The topological equivalence of one-dimensional basic sets of diffeomorphisms on two-dimensional manifolds, Uspehi Mat. Nauk 29, 6 (180) (1974) 163-164 MR 0440624 (55:13498)
  • 5. V. Grines, The topological conjugacy of diffeomorphisms of a two-dimensional manifold on one-dimensional orientable basic sets I, Trans. Moscow Math. Soc. 32 (1975) 31-56 MR 0418161 (54:6203)
  • 6. V. Grines, The topological conjugacy of diffeomorphisms of a two-dimensional manifold on one-dimensional orientable basic sets II, Trans. Moscow Math. Soc. 34 (1977) 243-252 MR 0474417 (57:14057)
  • 7. V. Grines, E. Zhuzhoma, Topological classification of orientable attractors on an $ n$-dimensional torus, Russian Math. Surveys 34 (1979) 163-164 MR 548425 (80k:58071)
  • 8. V. Grines, E. Zhuzhoma, On structurally stable diffeomorphisms with codimension one expanding attractors, Trans. AMS 357 (2005), no. 2, 617-667 MR 2095625 (2005k:37034)
  • 9. V. Grines, E. Zhuzhoma, Expanding attractors, Regular and chaotic dynamics 11, 2 (2006) 225-246 MR 2245079 (2007c:37015)
  • 10. A. Haefliger, Variétés feuilletés, Topologia Differenziale (Centro Int. Mat. Estivo, 1 deg Ciclo, Urbino (1962)). Lezione 2 Edizioni Cremonese, Rome 367-397 (1962) MR 0163327 (29:630)
  • 11. M. Hirsch, On invariant subsets of hyperbolic sets, Springer, New York (1970) 126-135 MR 0264684 (41:9275)
  • 12. M. Hirsch, C. Pugh, M. Shub, Invariant manifolds, Lecture Notes in Math. 583 (1977) MR 0501173 (58:18595)
  • 13. H. Kneser, Geschlossen Flächen in dreidimensionalen Mannigfaltigkeiten, Jahresbericht der Deutschen Mathematiker Vereinigung 38 (1929) 248-260
  • 14. R. Mañé, Expansive diffeomorphisms, Dyn. Syst., Proc. Symp. Univ. Warwick 1973/74, Lect. Notes Math. 468 (1975) 162-174. MR 0650658 (58:31263)
  • 15. R. Mañé, Quasi-Anosov diffeomorphisms and hyperbolic manifolds, Trans. AMS 229 (1977) 351-370. MR 0482849 (58:2894)
  • 16. R. Mañé, Invariant sets of Anosov diffeomorphisms, Invent. Math. 46 (1978), 147-152 MR 0488167 (58:7730)
  • 17. R. Mañé, A proof of the $ C^1$ stability conjecture, Publ. Math. Inst. Hautes Étud. Sci. 66 (1988) 161-210 MR 932138 (89e:58090)
  • 18. V. Medvedev, E. Zhuzhoma, On nonorientable two-dimensional basic sets on 3-manifolds, Sb. Math. 193 (2002), no. 5-6, 869-888 MR 1957954 (2004k:37040)
  • 19. J. Milnor, A unique decomposition theorem for 3-manifolds, Amer. J. of Math. 84, 1 (1962) 1-7 MR 0142125 (25:5518)
  • 20. R. Plykin, Hyperbolic attractors of diffeomorphisms, Russian Math. Surveys 35, 3 (1980) 109-121. MR 580625 (82k:58081)
  • 21. R. Plykin, Hyperbolic attractors of diffeomorphisms (the non-orientable case), Russian Math. Surveys 35, 4 (1980) 186-187. MR 0586207 (82k:58082)
  • 22. J. Rodriguez Hertz, R. Ures, J. Vieitez, On manifolds supporting quasi-Anosov diffeomorphisms, C.R. Acad. Sci. 334, 4, 321-323 (2002) MR 1891011 (2002m:37040)
  • 23. C. Robinson, Dynamical Systems Stability, Symbolic Dynamics, and Chaos, CRC Press, 1999. MR 1792240 (2001k:37003)
  • 24. R. Roussarie, Sur les feuilletages des variétés de dimension trois, Ann. Inst. Fourier 21, 3 (1971) 13-82. MR 0348768 (50:1263)
  • 25. M. Shub Global stability of dynamical systems, with the collab. of A. Fathi and R. Langevin. Transl. by J. Cristy, New York, Springer-Verlag. XII (1987). MR 869255 (87m:58086)
  • 26. S. Smale, Differentiable dynamical systems, Bull. AMS, 73, 6 (1967) 747-817. MR 0228014 (37:3598)
  • 27. S. Smale, The $ \Omega$-stability theorem, Global Analysis, Proc. Symp. Pure Math. 14 (1970) 289-297 MR 0271971 (42:6852)
  • 28. R. Williams. The `DA' maps of Smale and structural stability, Global Anal., Proc. Symp. Pure. Math., AMS 14 (1970), 329-334 MR 0264705 (41:9296)
  • 29. A. Zeghib, Subsystems of Anosov systems, Amer. J. Math. 117 (1995), 1431-1448 MR 1363074 (97e:58175)

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC (2000): 37D05, 37D20

Retrieve articles in all journals with MSC (2000): 37D05, 37D20


Additional Information

T. Fisher
Affiliation: Department of Mathematics, Brigham Young University, 292 TMCB, Provo, Utah 84602
Email: tfisher@math.byu.edu

M. Rodriguez Hertz
Affiliation: IMERL, Facultad de Ingeniería, University de la Republica, Julio Herrera y Reissig 565, 11300 Montevideo, Uruguay
Email: jana@fing.edu.uy

DOI: https://doi.org/10.1090/S0002-9947-09-04687-X
Keywords: Dynamical systems, hyperbolic set, robustly expansive, quasi-Anosov
Received by editor(s): May 8, 2007
Published electronically: February 10, 2009
Additional Notes: This work was partially supported by NSF Grant #DMS0240049, Fondo Clemente Estable 9021 and PDT
Article copyright: © Copyright 2009 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

American Mathematical Society