Ω-stable limit set explosions

Patterson, S. E.

doi:10.1090/S0002-9947-1986-0825737-9

$\Omega$-stable limit set explosions
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by S. E. Patterson

Trans. Amer. Math. Soc. 294 (1986), 775-798

DOI: https://doi.org/10.1090/S0002-9947-1986-0825737-9

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Abstract:

Certain diffeomorphisms of two-dimensional manifolds are considered. These diffeomorphisms have a finite hyperbolic limit set which contains a limit set cycle. The only nontransverse cycle connection in these cycles is a complete coincidence of one component of the unstable manifold of one periodic point with one component of the stable manifold of some other periodic point. A one-parameter family of diffeomorphisms containing the original diffeomorphism is described. It is shown that for parameter values arbitrarily near the parameter value corresponding to the original map these diffeomorphisms have a much enlarged limit set and are $\Omega$-stable.

References

Alan Dankner, Axiom A dynamical systems, cycles and stability, Topology 19 (1980), no. 2, 163–177. MR 572581, DOI 10.1016/0040-9383(80)90004-X
Sheldon E. Newhouse, Hyperbolic limit sets, Trans. Amer. Math. Soc. 167 (1972), 125–150. MR 295388, DOI 10.1090/S0002-9947-1972-0295388-6
S. Newhouse and J. Palis, Hyperbolic nonwandering sets on two-dimensional manifolds, Dynamical systems (Proc. Sympos., Univ. Bahia, Salvador, 1971) Academic Press, New York, 1973, pp. 293–301. MR 0339284
S. Newhouse and J. Palis, Cycles and bifurcation theory, Trois études en dynamique qualitative, Astérisque, No. 31, Soc. Math. France, Paris, 1976, pp. 43–140. MR 0516408
J. Palis, A note on $\Omega$-stability, Global Analysis (Proc. Sympos. Pure Math., Vols. XIV, XV, XVI, Berkeley, Calif., 1968) Amer. Math. Soc., Providence, R.I., 1970, pp. 221–222. MR 0270387
Charles C. Pugh, Russell B. Walker, and F. Wesley Wilson Jr., On Morse-Smale approximations–a counterexample, J. Differential Equations 23 (1977), no. 1, 173–182. MR 436218, DOI 10.1016/0022-0396(77)90140-1
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., Vols. XIV, XV, XVI, Berkeley, Calif., 1968) Amer. Math. Soc., Providence, R.I., 1970, pp. 289–297. MR 0271971