Bifurcation locus

Bifurcation locus

In complex dynamics, the bifurcation locus of a family of holomorphic functions informally is a locus of those maps for which the dynamical behavior changes drastically under a small perturbation of the parameter. Thus the bifurcation locus can be thought of as an analog of the Julia set in parameter space. Without doubt, the most famous example of a bifurcation locus is the boundary of the Mandelbrot set.

Parameters in the complement of the bifurcation locus are called J-stable.

References

[EL]:
Alexandre E. Eremenko and Mikhail Yu. Lyubich Dynamical properties of some classes of entire functions, Ann. Inst. Fourier (Grenoble) 42 (1992), no. 4, 989–1020, http://www.numdam.org/item?id=AIF_1992__42_4_989_0.
[L]:
Mikhail Yu. Lyubich, Some typical properties of the dynamics of rational mappings (Russian), Uspekhi Mat. Nauk 38 (1983), no. 5(233), 197–198.
[MSS]:
Ricardo Mañé, Paulo Sad and Dennis Sullivan, On the dynamics of rational maps, Ann. Sci. École Norm. Sup. (4) 16 (1983), no. 2, 193–217, http://www.numdam.org/item?id=ASENS_1983_4_16_2_193_0.
[McM]:
Curtis T. McMullen, Complex dynamics and renormalization, Annals of Mathematics Studies, 135, Princeton University Press, Princeton, NJ, 1994. ISBN 0-691-02982-2.

See also

Connectedness locus

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