Bifurcation currents in holomorphic dynamics on \bf P^k

2005/07/27 by Giovanni Bassanelli, Bassanelli, Giovanni, François Berteloot +1
Mathematics · #37F10 #37F45 #Complex Variables (math.CV) #Dynamical Systems (math.DS) #FOS: Mathematics #Geometric Analysis and Curvature Flows #Geometry and complex manifolds #Mathematical Dynamics and Fractals #math.CV #math.DS #msc:37F10 #msc:37F45

paper · pdf · doi:10.48550/arxiv.math/0507555

32 pages

openalex publication_date 2005/07/27 · arxiv created 2005/09/21 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

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 P¹) 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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