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Jumping stable manifolds for dissipative maps of the plane. (English) Zbl 0686.58008
An essential assumption of the authors is that basin boundaries arise ‘typically’ from stable manifolds of saddle points. Other separatrices are ignored. The ‘discontinuity’ argument is inspired by low accuracy numerical computations. A numerical study of a particular $${\mathbb{C}}\to {\mathbb{C}}$$ map, due to Ikeda, is used as confirming evidence. For $${\mathbb{C}}\to {\mathbb{C}}$$ maps it was already known to Julia (70 years ago) that basin boundaries can have an extremely complicated structure, not necessarily related to saddle-point manifolds. Other non-‘typical’ cases can be found in books, for example [in the reviewer and C. Mira: Recurrences and discrete dynamical systems (Lect. Notes Math. 809, 1980; Zbl 0449.58003)].
Reviewer: I.Gumowski

##### MSC:
 58D15 Manifolds of mappings 65Q05 Numerical methods for functional equations (MSC2000)
##### Keywords:
maps; basin boundaries; stable manifolds of saddle points
Full Text:
##### References:
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