Real saddle-node bifurcation from a complex viewpoint
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- by Michał Misiurewicz and Rodrigo A. Pérez PDF
- Conform. Geom. Dyn. 12 (2008), 97-108
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.References
- Núria Fagella and Antonio Garijo, The parameter planes of $\lambda z^m\exp (z)$ for $m\geq 2$, Comm. Math. Phys. 273 (2007), no. 3, 755–783. MR 2318864, DOI 10.1007/s00220-007-0265-8
- John Milnor, Remarks on iterated cubic maps, Experiment. Math. 1 (1992), no. 1, 5–24. MR 1181083
- MichałMisiurewicz and Ana Rodrigues, Double standard maps, Comm. Math. Phys. 273 (2007), no. 1, 37–65. MR 2308749, DOI 10.1007/s00220-007-0223-5
- Shizuo Nakane and Dierk Schleicher, On multicorns and unicorns. I. Antiholomorphic dynamics, hyperbolic components and real cubic polynomials, Internat. J. Bifur. Chaos Appl. Sci. Engrg. 13 (2003), no. 10, 2825–2844. MR 2020986, DOI 10.1142/S0218127403008259
- David Singer, Stable orbits and bifurcation of maps of the interval, SIAM J. Appl. Math. 35 (1978), no. 2, 260–267. MR 494306, DOI 10.1137/0135020
Additional Information
- Michał Misiurewicz
- Affiliation: Department of Mathematical Sciences, IUPUI, 402 N. Blackford Street, Indianapolis, Indiana 46202-3216
- MR Author ID: 125475
- 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
- 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. - © Copyright 2008 Michał Misiurewicz ; Rodrigo Pérez
- Journal: Conform. Geom. Dyn. 12 (2008), 97-108
- MSC (2000): Primary 37E05, 37H20, 37F99
- DOI: https://doi.org/10.1090/S1088-4173-08-00180-X
- MathSciNet review: 2425096