Dynamics on the Riemann Sphere: A Bodil Branner Festschrift

The papers collected in this volume were written to celebrate Bodil Branner’s 60th birthday. Most of them were presented at the 'Bodil Fest', a symposium on holomorphic dynamics held in June 2003 in Holbäk, Denmark. The main research theme of Bodil Branner is the iteration of cubic polynomials. Together with John H. Hubbard, she described the global topology of the parameter space C2 of polynomials of the form Pa,b(z) = z3 - a2z + b. Several decompositions of the parameter space have been considered. The first splitting is to separate the connectedness locus (where both critical points have bounded orbit and the Julia set is therefore connected) from the escape locus (where at least one critical point escapes to infinity). The second splitting is to foliate the escape locus into different hyper-surfaces, each one corresponding to a fixed maximal escape rate of the critical points. A particular way of constructing Teichmüller almost complex structures, which are invariant under Pa,b, was introduced as wringing and stretching of the complex structure; this technique is now referred to as Branner-Hubbard motion.
In the volume, Branner-Hubbard motion is described in the papers of C. L. Petersen, Tan Lei, and A. Douady. A survey paper of J. Milnor treats Lattès maps. A. Avila and M. Lyubich give examples of infinitely renormalizable quadratic maps whose Julia sets have Hausdorff dimension arbitrarily close to one. A. Chéritat studies the linearizability of the family Pθ(z) = e2πiθz + z2. Two papers (written by P. Blanchard et al. and P. Roesch) treat the parameter space of the family fλ(z) = z2 + λ/z2. T. Kawahira studies small perturbations of geometrically finite maps into other geometrically finite maps that are (semi)-conjugate on the Julia set to the original map. W. Jung constructs by quasi-conformal surgery a class of homeomorphisms of subsets of the Mandelbrot set. N. Fagella and Ch. Henriksen study Arnold tongues in the complexification of analytic diffeomorphisms fa,t(x) = x + t + (a/2π) sin(2πx) of the circle.

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