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Bifurcation Currents in Holomorphic Families of Rational Maps

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Pluripotential Theory

Part of the book series: Lecture Notes in Mathematics ((LNMCIME,volume 2075))

Abstract

These lectures are devoted to the study of bifurcations within holomorphic families of rational maps or polynomials by mean of ergodic and potential theoretic tools. After giving a general overview of the subject, we consider rational functions as ergodic dynamical systems and introduce the Green measure of a rational map and study the properties of its Lyapunov exponent. Next we consider Holomorphic families and introduce the class of hypersurfaces (Pern(w)) in the parameter space of a holomorphic family and study the connectedness locus in polynomial families. Then we introduce the bifurcation current and discuss equidistribution towards the bifurcation current and the self-intersection of the bifurcation current.

Je dédie ce texte à mes parents ainsi qu’à la mémoire de mon ami Giovanni Bassanelli.

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Acknowledgements

It is my pleasure to thank my colleagues Charles Favre, Thomas Gauthier and Nessim Sibony for their useful comments on the first draft of these notes. I also would like to thank the anonymous referee for having carefully read the manuscript and having helped me to improve it.

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  1. C.I.M.E Course, Cetraro, Italy

    François Berteloot

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Berteloot, F. (2013). Bifurcation Currents in Holomorphic Families of Rational Maps. In: Pluripotential Theory. Lecture Notes in Mathematics(), vol 2075. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-36421-1_1

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