Bifurcation Currents in Holomorphic Families of Rational Maps
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
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Je d´edie ce texte a` mes parents ainsi qu’`a la m´emoire de mon ami Giovanni Bassanelli.
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.
1 Introduction In these lectures we will study bifurcations within holomorphic families of polynomials or rational maps by mean of ergodic and pluripotential theoretic tools. A family of rational maps .f /2M , whose parameter space M is a complex manifold, is called a holomorphic family if the map .; z/ 7! f .z/ is holomorphic on M P1 and if the degree of f is constant on M . The simplest example is the quadratic polynomial family .z2 C /2C . The space of all rational maps of the same degree may also be considered as such a family.
F. Berteloot () C.I.M.E Course, Cetraro, Italy e-mail: [email protected] G. Patrizio et al., Pluripotential Theory, Lecture Notes in Mathematics 2075, DOI 10.1007/978-3-642-36421-1 1, © Springer-Verlag Berlin Heidelberg 2013
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F. Berteloot
The interest for bifurcations within holomorphic families of rational maps started in the eighties with the seminal works of Ma˜ne´ –Sad–Sullivan [42], Lyubich [41], and Douady–Hubbard [24, 25]. At the end of this decade, McMullen used Ma˜ne´ – Sad–Sullivan ideas and Thurston’s theory in his fundamental work on iterative rootfinding algorithms [45]. In any holomorphic family, the stability locus is the maximal open subset of the parameter space on which the Julia set moves continuously with the parameter. Its complement is called the bifurcation locus. In the quadratic polynomial family, the bifurcation locus is nothing but the boundary of the Mandelbrot set. Ma˜ne´ , Sad and Sullivan have shown that the stability locus is dense and their results also enlighted the (still open) question of the density of hyperbolic parameters. McMullen proved that any stable algebraic stable family of rational maps is either trivial or affine (i.e. consists of Latt`es examples), his classification of generally convergent algorithms follows from this central result. Potential theory has been introduced in the dynamical study of polynomials by Brolin in 1965. These tools and, more precisely the pluripotential theory developed after the fundamental works of Bedford–Taylor, turned out to be extremely powerful to study holomorphic dynamical systems depending on several complex variables. In this context, the compactness properties of cl
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