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The Mandelbrot set is universal. (English) Zbl 1062.37042
Lei, Tan, The Mandelbrot set, theme and variations. Cambridge: Cambridge University Press (ISBN 0-521-77476-4/pbk). Lond. Math. Soc. Lect. Note Ser. 274, 1-17 (2000).
From the introduction: Fix an integer $$d\geq 2$$, and let $$p_c(z)= z^d+ c$$. The generalized Mandelbrot set $$M_d\subset\mathbb{C}$$ is defined as the set of $$c$$ such that the Julia set $$J(p_c)$$ is connected. Equivalently, $$c\in M_d$$ iff $$p^n_c(0)$$ does not tend to infinity as $$n\to\infty$$. The traditional Mandelbrot set is the quadratic version $$M_2$$.
A holomorphic family of rational maps over $$X$$ is a holomorphic map $$f: X\times\widehat{\mathbb{C}}\to \widehat{\mathbb{C}}$$ where $$X$$ is a complex manifold and $$\widehat{\mathbb{C}}$$ is the Riemann sphere. For each $$t\in X$$ the family $$f$$ specializes to a rational map $$f_t:\widehat{\mathbb{C}}\to \widehat{\mathbb{C}}$$, denoted $$f_t(z)$$. For convenience we require that $$X$$ is connected and that $$\deg(f_t)\geq 2$$ for all $$t$$.
The bifurcation locus $$B(f)\subset X$$ is defined equivalently as the set of $$t$$ such that: (1) The number of attracting cycles of $$f_t$$ is not locally constant; (2) The period of the attracting cycles of $$f_t$$ is locally unbounded; or (3) The Julia set $$J(f_t)$$ does not move continuously (in the Hausdorff topology) over any neighborhood of $$t$$.
It is known that $$B(f)$$ is a closed, nowhere dense subset of $$X$$; its complement $$X- B(f)$$ is also called the $$J$$-stable set.
In this paper we show that every bifurcation set contains a copy of the boundary of the Mandelbrot set or its degree d generalization. The Mandelbrot sets $$M_d$$ are thus universal; they are initial objects in the category of bifurcations, providing a lower bound on the complexity of $$B(f)$$ for all families $$f_t$$.
For simplicity we first treat the case $$X= \Delta- \{t:|t|< 1\}$$.
For the entire collection see [Zbl 0935.00019].

##### MSC:
 37F45 Holomorphic families of dynamical systems; the Mandelbrot set; bifurcations (MSC2010) 37F10 Dynamics of complex polynomials, rational maps, entire and meromorphic functions; Fatou and Julia sets 30D05 Functional equations in the complex plane, iteration and composition of analytic functions of one complex variable