Displaying similar documents to “Stability and genericity in dynamical systems”

Prolongations and stability in dynamical systems

J. Auslander, P. Seibert (1964)

Annales de l'institut Fourier

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Les auteurs étudient la notion de prolongement au sens de T. Ura et ses relations avec la notion d’ensembles positivement invariants. La stabilité au sens de Liapounoff est équivalente à l’invariance par prolongement. Les auteurs dégagent ensuite la notion de “prolongements abstraits” et les notions de stabilité correspondantes; la stabilité absolue (associée au prolongement minimal transitif) et la stabilité asymptotique jouent un rôle important.

Ω-stability for maps with nonwandering critical points

J. Delgado, N. Romero, A. Rovella, F. Vilamajó (2007)

Fundamenta Mathematicae

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Sufficient conditions for a map having nonwandering critical points to be Ω-stable are introduced. It is not known if these conditions are necessary, but they are easily verified for all known examples of Ω-stable maps. Their necessity is shown in dimension two. Examples are given of Axiom A maps that have no cycles but are not Ω-stable.

C¹ stable maps: examples without saddles

J. Iglesias, A. Portela, A. Rovella (2010)

Fundamenta Mathematicae

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We give here the first examples of C¹ structurally stable maps on manifolds of dimension greater than two that are neither diffeomorphisms nor expanding. It is shown that an Axiom A endomorphism all of whose basic pieces are expanding or attracting is C¹ stable. A necessary condition for the existence of such examples is also given.

On absolute stability

Roger C. McCann (1972)

Annales de l'institut Fourier

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Absolute stability of a compact set is characterized by the cardinality of a fundamental system of positively invariant neighborhoods.