Bodil Branner

Homeomorphisms between limbs of the Mandelbrot set

Using a family of higher degree polynomials as a bridge, together with complex surgery techniques, we construct a homeomorphism between any two limbs of the Mandelbrot set of equal denominator. Induced by these homeomorphisms and complex conjugation, we obtain an involution between each limb and itself, whose fixed points form a topological arc. All these maps have counterparts at the combinatorial level relating corresponding external arguments. Assuming local connectivity of the Mandelbrot set we may conclude that the constructed homeomorphisms between limbs are compatible with the embeddings of the limbs in the plane. As usual we plough in the dynamical planes and harvest in the parameter space.

Classification of complex polynomial vector fields in one complex variable

This paper classifies the global structure of monic and centred one-variable complex polynomial vector fields. The classification is achieved by means of combinatorial and analytic data. More specifically, given a polynomial vector field, we construct a combinatorial invariant, describing the topology, and a set of analytic invariants, describing the geometry. Conversely, given admissible combinatorial and analytic data sets, we show using surgery the existence of a unique monic and centred polynomial vector field realizing the given invariants. This is the content of the Structure Theorem, the main result of the paper. This result is an extension and refinement of Douady et al. (Champs de vecteurs polynomiaux sur ℂ. Unpublished manuscript) classification of the structurally stable polynomial vector fields. We further review some general concepts for completeness and show that vector fields in the same combinatorial class have flows that are quasi-conformally equivalent.

THE PARAMETER SPACE FOR COMPLEX CUBIC POLYNOMIALS

Publisher Summary This chapter discusses the parameter space for complex cubic polynomials. The dynamical behavior under iteration of a rational map is dominated by the behavior of the critical points. A rough partition of the parameter space can be based on the possible behaviors of the critical points. The main tools for understanding iteration of a monic polynomial P are the ϕ p -map and the h p -map: the ϕ p -map is defined in a neighborhood of ∞. The chapter describes dichotomy for dynamical behavior. Each copy of the Mandelbrot set corresponds to a Cantor set construction in the set of angles for rays. Cantor sets live in the dynamical planes and in the parameter space.

Iterations by Odd functions with two extrema

Abstract Let I = [−1, 1] and fI → I be continuous, piecewise monotone and odd with two extrema. A periodic orbit is called symmetric if − x is in the orbit when x is in the orbit. A periodic orbit which is not symmetric is called asymmetric. The first result of this paper proves an ordering of the periods for the symmetric orbits. There are two possibilities depending on how f behaves in a neighbourhood of 0. The second result of this paper proves that for a one-parameter family of odd functions with negative Schwarzian derivative there are three different types of nondegenerate bifurcations: saddle node, period-doubling pitchfork and period-preserving pitchfork. The last type of bifurcation occurs exactly when a symmetric orbit bifurcates to two asymmetric orbits.

Extensions of homeomorphisms between limbs of the Mandelbrot set

Using holomorphic surgery techniques, we construct a homeomorphism between a neighborhood of any limb without root point of the Mandelbrot set and a neighborhood of any other of equal denominator, in such a way that the limbs are mapped to each other. On the limbs, the homeomorphism coincides with that constructed in “Homeomorphisms between limbs of the Mandelbrot set” (J. Geom. Anal. 9 (1999), 327–390) which proves – without assuming local connectivity of the Mandelbrot set – that these maps are compatible with the embedding of the limbs in the plane. Outside the limbs, the constructed extension is quasi-conformal.

Cover illustration: The limit elephant

This years cover illustration, reproduced as figure 4, comes from the branch of mathematics called complex dynamics. It shows a subset of the parameter space of a certain family of maps.