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)

 
 

 

Markov neighborhoods for zero-dimensional basic sets


Author: Dennis Pixton
Journal: Trans. Amer. Math. Soc. 279 (1983), 431-462
MSC: Primary 58F10; Secondary 58F18
DOI: https://doi.org/10.1090/S0002-9947-1983-0709562-2
MathSciNet review: 709562
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: We extend the local stable and unstable laminations for a zero-dimensional basic set to semi-invariant laminations of a neighborhood, and use these extensions to construct the appropriate analog of a Markov partition, which we call a Markov neighborhood. The main applications we give are in the perturbation theory for stable and unstable manifolds; in particular, we prove a transversality theorem. For these applications we require not only that the basic sets be zero dimensional but that they satisfy certain tameness assumptions. This leads to global results on improving stability properties via small isotopies.


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

  • [B] R. Bowen, Equilibrium states and the ergodic theory of Anosov diffeomorphisms, Chap. 3, Lecture Notes in Math., vol. 470, Springer-Verlag, Berlin, 1975. MR 0442989 (56:1364)
  • [BF] R. Bowen and J. Franks, Homology for zero-dimensional basic sets, Ann. of Math. (2) 106 (1977), 73-92. MR 0458492 (56:16692)
  • [D] J. Dugundji, An extension of Tietze's theorem, Pacific J. Math. 1 (1951), 353-367. MR 0044116 (13:373c)
  • [FR] J. Franks and C. Robinson, A quasi-Anosov diffeomorphism that is not Anosov, Trans. Amer. Math. Soc. 223 (1976), 267-278. MR 0423420 (54:11399)
  • [H] M. Hirsch, Differential topology, Graduate Texts in Math., no. 33, Springer-Verlag, New York, 1976. MR 0448362 (56:6669)
  • [HP] M. Hirsch and C. Pugh, Stable manifolds and hyperbolic sets, Proc. Sympos. Pure Math., vol. 14, Amer. Math. Soc., Providence, R.I., 1970, pp. 133-164. MR 0271991 (42:6872)
  • [HPS] M. Hirsch, C. Pugh and M. Shub, Invariant manifolds, Lecture Notes in Math., vol. 583, Springer-Verlag, Berlin, 1977. MR 0501173 (58:18595)
  • [HPPS] M. Hirsch, J. Palis, C. Pugh and M. Shub, Neighborhoods of hyperbolic sets, Invent. Math. 9 (1970), 121-134. MR 0262627 (41:7232)
  • [K] L. Keldys, Topological embeddings in Euclidean space, Proc. Steklov Inst. Math. 81 (1966). MR 0232371 (38:696)
  • [M] W. de Melo, Structural stability of diffeomorphisms on two-manifolds, Invent. Math. 21 (1973), 233-246. MR 0339277 (49:4037)
  • [N] S. Newhouse, On simple arcs between structurally stable flows, Dynamical Systems--Warwick 1974, Lecture Notes in Math., vol. 468, Springer-Verlag, Berlin, 1975, pp. 209-233. MR 0650638 (58:31249)
  • [N1] -, Nondensity of Axiom $ A(a)$ on $ S^2$, Proc. Sympos. Pure Math., vol. 14, Amer. Math. Soc., Providence, R. I., 1970, pp. 191-202. MR 0277005 (43:2742)
  • [N2] -, The abundance of wild hyperbolic sets and non-smooth stable sets for diffeomorphisms, Inst. Hautes Études Sci. Publ. Math. 50 (1979), 101-152.
  • [PS] J. Palis and S. Smale, Structural stability theorems, Proc. Sympos. Pure Math., vol. 14, Amer. Math. Soc., Providence, R. I., 1970, pp. 223-231. MR 0267603 (42:2505)
  • [Ro] J. Robbin, A structural stability theorem, Ann. of Math. (2) 94 (1971), 447-493. MR 0287580 (44:4783)
  • [R] C. Robinson, Structural stability of $ {C^1}$ diffeomorphisms, J. Differential Equations 22 (1976), 28-73. MR 0474411 (57:14051)
  • [RW] C. Robinson and R. F. Williams, Classification of expanding attractors: an example, Topology 15 (1976), 321-323. MR 0415682 (54:3762)
  • [SS] M. Shub and D. Sullivan, Homology theory and dynamical systems, Topology 14 (1975), 109-132. MR 0400306 (53:4141)
  • [S] S. Smale, Differentiable dynamical systems, Bull. Amer. Math. Soc. 73 (1967), 747-817. MR 0228014 (37:3598)
  • [S1] -, The $ \Omega $-stability theorem, Proc. Sympos. Pure Math., vol. 14, Amer. Math. Soc., Providence, R.I., 1970, pp. 289-298. MR 0271971 (42:6852)
  • [S2] -, Stability and isotopy in discrete dynamical systems, Symposium on Dynamical Systems-- Salvador, Academic Press, New York, 1973, pp. 527-530. MR 0339222 (49:3984)
  • [St] J. Stallings, The piecewise-linear structure of Euclidean space, Proc. Cambridge Philos. Soc. 58 (1962), 481-488. MR 0149457 (26:6945)
  • [Z] E. C. Zeeman, $ {C^0}$ density of stable diffeomorphisms and flows, Univ. of Warwick, preprint, 1973.

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC: 58F10, 58F18

Retrieve articles in all journals with MSC: 58F10, 58F18


Additional Information

DOI: https://doi.org/10.1090/S0002-9947-1983-0709562-2
Keywords: Structural stability, Axiom $ A$, basic set, lamination, Markov partition
Article copyright: © Copyright 1983 American Mathematical Society

American Mathematical Society