Transcendental dynamics

Entire functions (holomorphic maps  f:\mathbb{C} \to \mathbb{C}) come in two kinds: algebraic and transcendental. Maps of the first kind are polynomials and have finite degrees, so they extend to the Riemann sphere, and their superattracting fixed point at  \infty makes them the simplest holomorphic maps to study (even though they still provide serious challenges!). On the other hand, transcendental entire functions have essential singularities at  \infty, and their study is much harder. One of the most important tools of polynomial dynamics, dynamic rays, are defined quite easily using local coordinates around  \infty, and it had been an open question whether there is a similar structure in the transcendental case. In general, the answer is negative, but we identified a large class of transcendental entire functions for which dynamic rays can be defined (essentially as path components of escaping points: points that converge to  \infty under iteration).

Many mysterious features occur in transcendental dynamics. For instance, Karpińska discovered the following paradox: in certain cases (such as  z\mapsto \lambda e^z with  \lambda \in (0,1/e)), the union of all dynamic rays has Hausdorff dimension 1, but their endpoints (a single point at the end of each ray) have dimension 2. We extended this result for other maps (such as  \pi \sin z) to show that the endpoints even may have infinite 2-dimensional measure, and in fact, be the complement of the 1-dimensional set of rays.

Transcendental dynamics is a very interesting, sometimes counterintuitive field of study, with a very lively and active development, especially in recent years.

 

Some of our main results:

  • Construction of dynamic rays for a large class of transcendental entire functions (and disproof of Eremenko’s conjecture that such rays might always exist):
    Günter Rottenfußer, Johannes Rückert, Lasse Rempe, and Dierk Schleicher: Dynamic rays of bounded-type entire functions. Annals of Mathematics, 173 1 (2011), 77-125. (arxiv)
  • An extension to Thurston’s characterization theorem of rational maps to a family of transcendental maps:
    John Hubbard, Dierk Schleicher, and Mitsuhiro Shishikura: Exponential Thurston maps and limits of quadratic differentials. Journal of the American Mathematical Society, 22 (2009), 77-117. (article)
  • A systematic study of the bifurcation locus of exponential maps, including a proof of a conjecture by Baker-Rippon, Eremenko-Lyubich, and Devaney-Goldberg-Hubbard:
    Lasse Rempe and Dierk Schleicher: Bifurcations in the space of exponential maps. Inventiones Mathematicae 175 (2009), 103-135. (arxiv)
  • A detailed study of the dynamics of postcritically finite maps in the sine family, including the result that every point in \mathbb{C} is either on a unique dynamic ray or the landing point of one or several dynamic rays, with the strongest possible version of Karpinska’s dimension paradox
    Dierk Schleicher: The dynamical fine structure of iterated cosine maps and a dimension paradox. Duke Mathematics Journal 136 2 (2007), 343-356. (arxiv)
    Dierk Schleicher: Hausdorff dimension, its properties, and its surprises. American Mathematical Monthly, 114 6 (2007), 509-528. (arxiv)

Experts within our team: Saikat Batabyal, Bayani Hazemach, Dierk Schleicher