Bifurcation locus

From Wikipedia, the free encyclopedia

In complex dynamics, the bifurcation locus of a family of holomorphic functions informally consists 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.

[edit] 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.