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.

ee also

Connectedness locus


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