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Conformal Geometry and Dynamics
Conformal Geometry and Dynamics
ISSN 1088-4173

Real saddle-node bifurcation from a complex viewpoint


Authors: Michał Misiurewicz and Rodrigo A. Pérez
Journal: Conform. Geom. Dyn. 12 (2008), 97-108
MSC (2000): Primary 37E05, 37H20, 37F99
Published electronically: July 21, 2008
MathSciNet review: 2425096
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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: http://dx.doi.org/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
Published electronically: 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.
Article copyright: © Copyright 2008 Micha\l Misiurewicz \and Rodrigo P\'erez