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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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