Markov neighborhoods for zero-dimensional basic sets
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- by Dennis Pixton PDF
- Trans. Amer. Math. Soc. 279 (1983), 431-462 Request permission
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
- Rufus Bowen, Equilibrium states and the ergodic theory of Anosov diffeomorphisms, Lecture Notes in Mathematics, Vol. 470, Springer-Verlag, Berlin-New York, 1975. MR 0442989
- Rufus Bowen and John Franks, Homology for zero-dimensional nonwandering sets, Ann. of Math. (2) 106 (1977), no. 1, 73–92. MR 458492, DOI 10.2307/1971159
- J. Dugundji, An extension of Tietze’s theorem, Pacific J. Math. 1 (1951), 353–367. MR 44116
- John Franks and Clark Robinson, A quasi-Anosov diffeomorphism that is not Anosov, Trans. Amer. Math. Soc. 223 (1976), 267–278. MR 423420, DOI 10.1090/S0002-9947-1976-0423420-9
- Morris W. Hirsch, Differential topology, Graduate Texts in Mathematics, No. 33, Springer-Verlag, New York-Heidelberg, 1976. MR 0448362
- Morris W. Hirsch and Charles C. Pugh, Stable manifolds and hyperbolic sets, Global Analysis (Proc. Sympos. Pure Math., Vol. XIV, Berkeley, Calif., 1968) Amer. Math. Soc., Providence, R.I., 1970, pp. 133–163. MR 0271991
- M. W. Hirsch, C. C. Pugh, and M. Shub, Invariant manifolds, Lecture Notes in Mathematics, Vol. 583, Springer-Verlag, Berlin-New York, 1977. MR 0501173
- M. Hirsch, J. Palis, C. Pugh, and M. Shub, Neighborhoods of hyperbolic sets, Invent. Math. 9 (1969/70), 121–134. MR 262627, DOI 10.1007/BF01404552
- L. V. Keldyš, Topological imbeddings in Euclidean space, Proceedings of the Steklov Institute of Mathematics, No. 81 (1966), American Mathematical Society, Providence, R.I., 1968. Translated from the Russian by J. Zilber. MR 0232371
- W. de Melo, Structural stability of diffeomorphisms on two-manifolds, Invent. Math. 21 (1973), 233–246. MR 339277, DOI 10.1007/BF01390199
- S. Newhouse, On simple arcs between structurally stable flows, Dynamical systems—Warwick 1974 (Proc. Sympos. Appl. Topology and Dynamical Systems, Univ. Warwick, Coventry, 1973/1974; presented to E. C. Zeeman on his fiftieth birthday), Lecture Notes in Math., Vol. 468, Springer, Berlin, 1975, pp. 209–233. MR 0650638
- Sheldon E. Newhouse, Nondensity of axiom $\textrm {A}(\textrm {a})$ on $S^{2}$, Global Analysis (Proc. Sympos. Pure Math., Vol. XIV, Berkeley, Calif., 1968) Amer. Math. Soc., Providence, R.I., 1970, pp. 191–202. MR 0277005 —, The abundance of wild hyperbolic sets and non-smooth stable sets for diffeomorphisms, Inst. Hautes Études Sci. Publ. Math. 50 (1979), 101-152.
- J. Palis and S. Smale, Structural stability theorems, Global Analysis (Proc. Sympos. Pure Math., Vol. XIV, Berkeley, Calif., 1968) Amer. Math. Soc., Providence, R.I., 1970, pp. 223–231. MR 0267603
- J. W. Robbin, A structural stability theorem, Ann. of Math. (2) 94 (1971), 447–493. MR 287580, DOI 10.2307/1970766
- Clark Robinson, Structural stability of $C^{1}$ diffeomorphisms, J. Differential Equations 22 (1976), no. 1, 28–73. MR 474411, DOI 10.1016/0022-0396(76)90004-8
- Clark Robinson and Robert Williams, Classification of expanding attractors: an example, Topology 15 (1976), no. 4, 321–323. MR 415682, DOI 10.1016/0040-9383(76)90024-0
- M. Shub and D. Sullivan, Homology theory and dynamical systems, Topology 14 (1975), 109–132. MR 400306, DOI 10.1016/0040-9383(75)90022-1
- S. Smale, Differentiable dynamical systems, Bull. Amer. Math. Soc. 73 (1967), 747–817. MR 228014, DOI 10.1090/S0002-9904-1967-11798-1
- S. Smale, The $\Omega$-stability theorem, Global Analysis (Proc. Sympos. Pure Math., Vol. XIV, Berkeley, Calif., 1968) Amer. Math. Soc., Providence, R.I., 1970, pp. 289–297. MR 0271971
- Steve Smale, Stability and isotopy in discrete dynamical systems, Dynamical systems (Proc. Sympos., Univ. Bahia, Salvador, 1971) Academic Press, New York, 1973, pp. 527–530. MR 0339222
- John Stallings, The piecewise-linear structure of Euclidean space, Proc. Cambridge Philos. Soc. 58 (1962), 481–488. MR 149457 E. C. Zeeman, ${C^0}$ density of stable diffeomorphisms and flows, Univ. of Warwick, preprint, 1973.
Additional Information
- © Copyright 1983 American Mathematical Society
- 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