×

zbMATH — the first resource for mathematics

The abundance of wild hyperbolic sets and non-smooth stable sets for diffeomorphisms. (English) Zbl 0445.58022

MSC:
37D99 Dynamical systems with hyperbolic behavior
37D15 Morse-Smale systems
37C75 Stability theory for smooth dynamical systems
PDF BibTeX XML Cite
Full Text: DOI Numdam EuDML
References:
[1] R. Abraham andS. Smale, Non-genericity of \(\Omega\)-stability,Proc. A.M.S. Symp. in Pure Math.,14 (1970), 5–9.
[2] D. Anosov, Geodesic flows on closed Riemann manifolds with negative curvature,Proc. Steklov Inst. of Math., no 90 (1967). · Zbl 0176.19101
[3] R. Bowen, Topological entropy and Axiom A,Proc. A.M.S. Symp. in Pure Math.,14 (1970), 23–42.
[4] R. Bowen, Equilibrium states and the ergodic theory of Anosov diffeomorphisms,Lecture Notes in Math.,470, Berlin - Heidelberg - New York, Springer-Verlag, 1975. · Zbl 0308.28010
[5] R. Bowen, A Horseshoe with positive measure,Inventiones Math.,29 (1975), 203–204. · Zbl 0306.58013 · doi:10.1007/BF01389849
[6] R. Bowen andO. Landford, Zeta functions of restrictions of the shift transformations,Proc. A.M.S. Symp. Pure Math.,14 (1970), 43–51.
[7] P. Fatou, Sur les équations fonctionnelles,Bull. Soc. Math. France,47 (1919), 161–271, et48 (1920), 33–94 et 208–314. · JFM 47.0921.02
[8] M. Hénon, A two-dimensional mapping with a strange attractor,Comm. Math. Phys.,50 (1976), 69–78. · Zbl 0576.58018 · doi:10.1007/BF01608556
[9] M. Hirsch andC. Pugh, Stable manifolds and hyperbolic sets,Proc. A.M.S. Symp. in Pure Math.,14 (1970), 133–163. · Zbl 0215.53001
[10] M. Hirsch, J. Palis, C. Pugh andM. Shub, Neighborhoods of hyperbolic sets,Inventiones Math.,9 (1970), 121–134. · Zbl 0191.21701 · doi:10.1007/BF01404552
[11] M. Hirsch, C. Pugh andM. Shub, Invariant manifolds,Lecture Notes in Math.,583, Berlin - Heidelberg - New York, Springer Verlag, 1978. · Zbl 0355.58009
[12] C. Hayashi andY. Ueda, Behavior of solutions for certain types of non-linear differential equations of the second order,Non-linear Vibration Problems, Zagadnienia Drgan Nieliniowych,14 (1973), 341–351. · Zbl 0354.34039
[13] J. Moser, Stable and random motions in dynamical systems,Ann. of Math. Studies,77, Princeton, N.J., Princeton Uńiv. Press, 1973. · Zbl 0271.70009
[14] S. Newhouse, Non-density of Axiom A(a) on Sr,Proc. A.M.S. Symp. in Pure Math.,14 (1970), 191–203.
[15] S. Newhouse, Hyperbolic limit sets,Trans. Amer. Math. Soc.,167 (1972), 125–150. · Zbl 0239.58009 · doi:10.1090/S0002-9947-1972-0295388-6
[16] S. Newhouse, Diffeomorphisms with infinitely many sinks,Topology,13 (1974), 9–18. · Zbl 0275.58016 · doi:10.1016/0040-9383(74)90034-2
[17] S. Newhouse, Lectures on Dynamical Systems,C.I.M.E. Summer School in Dynamical Systems, 1978.
[18] S. Newhouse, Conservative systems and two problems of Smale,Lecture Notes in Math.,525, Berlin - Heidelberg - New York, Springer Verlag, 1975. · Zbl 0307.58010
[19] S. Newhouse andJ. Palis, Bifurcations of Morse-Smale dynamical systems,Dynamical Systems, ed. M. Peixoto, N.Y., Academic Press, 1973, 303–366.
[20] S. Newhouse andJ. Palis, Cycles and Bifurcations theory,Astérisque,31, 1976, 43–141.
[21] M. Shub andD. Sullivan, Homology theory and dynamical systems,Topology,14 (1975), 109–132. · Zbl 0408.58023 · doi:10.1016/0040-9383(75)90022-1
[22] C. Simon,Thesis, Northwestern University, 1970.
[23] S. Smale, Diffeomorphisms with many periodic points,Differential and Combinatorial Topology, Princeton, 1965, 63–80. · Zbl 0142.41103
[24] S. Smale, Differentiable dynamical systems,Bull. A.M.S.,73 (1967), 747–817. · Zbl 0202.55202 · doi:10.1090/S0002-9904-1967-11798-1
[25] S. Smale, Stability and isotopy in discrete dynamical systems,Dynamical systems, ed. M. Peixoto, N.Y., Academic Press, 1973, 527–530. · Zbl 0284.58013
[26] S. Smale, Notes on dynamical systems,Proc. A.M.S. Symp. in Pure Math.,14 (1970), 277–289. · Zbl 0205.54201
[27] P. Plikin, Sources and sinks in Axiom A Diffeomorphisms on surfaces,Mat. Sbornik,94 (136), no 2 (6), 1974, 243–264 (Russian).
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.