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].

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 |