Real saddle-node bifurcation from a complex viewpoint

Misiurewicz, Michał; Pérez, Rodrigo

doi:10.1090/S1088-4173-08-00180-X

Real saddle-node bifurcation from a complex viewpoint
HTML articles powered by AMS MathViewer

by Michał Misiurewicz and Rodrigo A. Pérez

Conform. Geom. Dyn. 12 (2008), 97-108

DOI: https://doi.org/10.1090/S1088-4173-08-00180-X

Published electronically: July 21, 2008

PDF

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