Parameter spaces and the Mandelbrot set

Within holomorphic dynamics, much of the deepest pioneering work has been done in the ground-breaking context of iterated quadratic polynomials and its parameter space, the Mandelbrot set \mathscr{M}, with the promise that the fundamental results would be relevant in more general context. Especially in recent years, this hope has seen significant progress, with substantial results on large spaces of iterated holomorphic maps.

Our research is embedded in this successful context. It starts with a study of a fundamental issue of the Mandelbrot set, working towards the question of local connectivity, and carries over methods and results to spaces of higher degree polynomials and ultimately to selected transcendental maps. The underlying goal is known as combinatorial rigidity: can any two holomorphic dynamical systems be distinguished in purely combinatorial terms? It turns out that for quadratic polynomials this question is equivalent to a good understanding of the topology of the Mandelbrot set: combinatorial rigidity of quadratic polynomials is equivalent to the statement that the Mandelbrot set is locally connected (which really means that we have a good topological model for \mathscr{M}).

This issue has been pioneered by Adrien Douady and John Hubbard since the 1980’s, and then substantially refined by Jean-Christophe Yoccoz in the 1990’s and by Mikhail Lyubich since then. Currently, all quadratic polynomials are now known to be combinatorially rigid, except certain ones that are infinitely renormalizable: this is one of the reasons why renormalization theory is important in holomorphic dynamics.

While the Mandelbrot set still serves as the prototypical parameter space in holomorphic dynamics that attracts serious studies, one direction of research activities is to investigate more general parameter spaces, often inspired by our knowledge of the Mandelbrot set. For instance, McMullen showed “the Mandelbrot set is universal”: every non-trivial space of holomorphic maps  f_\lambda that depends holomorphically on a complex variable  \lambda contains, in every neighborhood of every point in the “bifurcation locus”, infinitely many homeomorphic copies of \mathscr{M}.

Recently, we also completed a study of the tricorn, the connectedness locus of complex antiholomorphic quadratic polynomials: this tricorn was discovered, among other people, by John Milnor in his study of iterated real cubic polynomials.

 

Some of our main results:

  • A simplified description of the structure of the Mandelbrot set:
    Dierk Schleicher: Rational Parameter Rays of the Mandelbrot Set. Astérisque 261 (2000), 409-447. (article)
    Dierk Schleicher: On fibers and local connectivity of Mandelbrot and Multibrot sets. In: M. Lapidus, M. van Frankenhuysen (eds): Fractal Geometry and Applications: A Jubilee of Benoit Mandelbrot. Proc. of Simp. in Pure Math. 72, Amer. Math. Soc. (2004), 477–507. (article)
  • A systematic study of the space of antiholomorphic quadratic polynomials and the tricorn:
    Hiroyuki Inou, Sabyasachi Mukherjee: Non-landing parameter rays of the multicorns. Inventiones Mathematicae, 202 (2015), 1-25.  (arxiv)
    Sabyasachi Mukherjee, Shizuo Nakane, and Dierk Schleicher: On Multicorns and Unicorns II: bifurcations in spaces of antiholomorphic polynomials. Ergodic Theory and Dynamical Systems (to appear)(arxiv)
    Sabyasachi Mukherjee: Antiholomorphic Dynamics: Topology of Parameter Spaces and Discontinuity of Straightening, (2015). (thesis)
  • A study of the bifurcation locus of exponential maps, in analogy (and with interesting differences) to the Mandelbrot set:
    Lasse Rempe and Dierk Schleicher: Bifurcations in the space of exponential maps. Inventiones Mathematicae 175 (2009), 103-135. (arxiv)
  • A recent study of parameter spaces of rational maps that arise as Newton maps:
    Russell Lodge, Yauhen “Zhenya” Mikulich, and Dierk Schleicher: Combinatorial properties of Newton maps. (arxiv)
    Russell Lodge, Yauhen “Zhenya” Mikulich, and Dierk Schleicher: A classification of postcritically finite Newton maps. Submitted. (arxiv)

Experts within our team: Dima Dudko, Wolf Jung, Dierk Schleicher.