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Bifurcation currents in holomorphic dynamics on ${\bf P}^k$

Giovanni Bassanelli·François Berteloot
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Abstract

We establish a formula for the sum of the Lyapounov exponents of an holomorphic endomorphism of ${\bf P}^k$. For an holomorphic family of such endomorphisms we define the {\em bifurcation current} as $dd^cL$ and show that it vanishes when the repulsive cycles move holomorphically. We then prove a formula which relates this current with the interaction between the Green current and the current of integration on the critical set. In the 1-dimensional case (i.e. for ${\bf P}^1$) we find a geometrical description of the support of this current and its powers. Finally we introduce the {\em bifurcation measure} giving some applications. This last part may be interpreted as a generalization of Mane-Sad-Sullivan theory based on pluri-potentialist methods.

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Giovanni Bassanelli, François Berteloot. 2005-09-21. Bifurcation currents in holomorphic dynamics on ${\bf P}^k$. https://arxiv.org/abs/math/0507555

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