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Real saddle-node bifurcation from a complex viewpoint

Author(s): Michał Misiurewicz; Rodrigo A. Pérez
Journal: Conform. Geom. Dyn. 12 (2008), 97-108.
MSC (2000): Primary 37E05, 37H20, 37F99
Posted: July 21, 2008
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Abstract: During a saddle-node bifurcation for real analytic interval maps, a pair of fixed points, attracting and repelling, collide and disappear. From the complex point of view, they do not disappear, but just become complex conjugate. The question is whether those new complex fixed points are attracting or repelling. We prove that this depends on the Schwarzian derivative $ S$ at the bifurcating fixed point. If $ S$ is positive, both fixed points are attracting; if it is negative, they are repelling.

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Additional Information:

Michał Misiurewicz
Affiliation: Department of Mathematical Sciences, IUPUI, 402 N. Blackford Street, Indianapolis, Indiana 46202-3216
Email: mmisiure@math.iupui.edu

Rodrigo A. Pérez
Affiliation: Department of Mathematical Sciences, IUPUI, 402 N. Blackford Street, Indianapolis, Indiana 46202-3216
Email: rperez@math.iupui.edu

DOI: 10.1090/S1088-4173-08-00180-X
PII: S 1088-4173(08)00180-X
Keywords: Saddle-node, Schwarzian derivative, parabolic point
Received by editor(s): December 12, 2007
Posted: July 21, 2008
Additional Notes: The first author was partially supported by NSF grant DMS 0456526
The second author was partially supported by NSF grant DMS 0701557.
Copyright of article: Copyright 2008, Micha\l Misiurewicz \and Rodrigo P\'erez

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