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Higher bifurcation currents, neutral cycles and the Mandelbrot set. (English) Zbl 1325.37027
Let $$f$$ be a rational map. It is well known that $$f$$ admits a unique maximal entropy measure $$\mu_f$$ on the Riemann sphere $$\mathbb{P}^1$$. Let $$(f_{\lambda})_{\lambda \in \Lambda}$$ be a holomorphic family of degree-$$d$$ rational maps. The bifurcation current of the family $$(f_{\lambda})_{\lambda \in \Lambda}$$ (or of the moduli space $${\mathcal{M}}_d$$ of degree $$d$$-rational maps) is the closed positive $$(1,1)$$-current on $$\Lambda$$ defined by $$T_{{\text{bif}}}=dd^cL$$, where, for any rational map $$f$$ of degree $$d \geq 2$$, $L(f)=\int_{\mathbb{P}^1}\log(|f'|) d\mu_f,$ $$L(f)$$ is the Lyapunov exponent of $$f$$.
In this paper, the author generalizes the McMullen’s theorem on the universality of the Mandelbrot set and proves the following theorem.
Theorem. Let $$T_{{\text{bif}}}$$ be the bifurcation current of the moduli space $${\mathcal{M}}_d$$ of degree $$d$$-rational maps. For any $$1 \leq k \leq 2d-2$$ and any $$\Theta_k=(\theta_1,\dots,\theta_k) \in (\mathbb{R}/\mathbb{Z})^k$$, we have $${\text{supp}}(T_{{\text{bif}}}^k)={\overline{\mathcal{Z}_k(\Theta_k)}}=\overline{\text{Prerep}(k)}$$, where $\text{Prerep}(k)=\big\{[f] \in {\mathcal{M}}_d; f \text{ has } k \text{ critical points preperiodic to repelling cycles}\big\},$ and $\mathcal{Z}_k(\Theta_k)=\big\{[f] \in {\mathcal{M}}_d; f \text{ has } k \text{ distinct cycles of resp. multipliers } e^{2i\pi\theta_1},\dots,e^{2i\pi\theta_k}\big\},$

##### MSC:
 37F45 Holomorphic families of dynamical systems; the Mandelbrot set; bifurcations (MSC2010) 32U15 General pluripotential theory 28A78 Hausdorff and packing measures
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