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Bifurcation currents in holomorphic dynamics on $$\mathbb P^k$$. (English) Zbl 1136.37025
Authors’ abstract: We use pluri-potential theory to study the bifurcations of holomorphic families $$\{f_\lambda\}_{\lambda\in X}$$ of rational maps on $$\mathbb{P}^1$$ or endomorphisms of $$\mathbb{P}^k$$. To this purpose we establish some formulas for $$L(f_\lambda)$$ and $$dd^cL(f_\lambda)$$ where $$L(f_\lambda)$$ is the sum of the Lyapunov exponents of $$f_\lambda$$ with respect to the maximal entropy measure. We show that the bifurcation current $$dd^cL(f_\lambda)$$ both detects the instability of repulsive cycles and the interaction between critical and Julia sets. For families of rational maps of degree $$d$$, we introduce a bifurcation measure defined by $$(dd^cL(f_\lambda))^{2d-2}$$ and study its first properties. In particular, we show that the support of this measure is contained in the closure of the set of rational maps having $$2d-2$$ distinct Cremer-cycles. This approach yields to a purely potential-theoretic proof of the Mané-Sad-Sullivan theorem and, moreover, allows us to extend it.
Reviewer: Pei-Chu Hu (Jinan)

##### MSC:
 37F10 Dynamics of complex polynomials, rational maps, entire and meromorphic functions; Fatou and Julia sets
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