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

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.

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