Symbolic dynamics and core entropy
One of the particularly attractive and powerful features of holomorphic dynamics is the fruitful interaction between the rich and surprising versatility of dynamical systems and the stiffnes provided by complex analysis. Many difficult issues that are beyond hope in general dynamics have deep and beautiful answers in holomorphic dynamics, for instance Yoccoz' solution of the notorious "small divisors" problem from physics for quadratic polynomials and other families of holomorphic maps. In many cases, the underlying idea is that questions from dynamics can be translated to issues of symbolic dynamics that often yield surprisingly simple and complete answers to the original problem.
As so often, one prototypical case is the Mandelbrot set: different quadratic polynomials can not only be distinguished by their complex parameters, but by combinatorial invariants such as external angles or kneading sequences (at least subject to the famous MLC conjecture: the Mandelbrot is locally connected). Every quadratic polynomial has such an associated combinatorial invariant, and studying the structure of the Mandelbrot set to a large extent means studying the resulting combinatorial invariants: for external angles, every angle does occur and the interesting question is which angles describe the same dynamics; for kneading sequence, the relevant question is which kneading sequences occur for complex quadratic polynomials. These invariants lead to combinatorial models of the Julia sets and the Mandelbrot set in terms of Thurston's laminations or Douady's pinched disk models, and we know that these are topologically true models when the Julia sets or the Mandelbrot set are locally connected.
While a lot is known about the combinatorial structure of the Mandelbrot set, much less is known already in the case of cubic polynomials, and even more interesting questions arise in transcencental dynamics.
Symbolic dynamics also plays an important role in the context of Thurston theory: given postcritically finite rational maps (or postsingularly finite transcendental maps), one has to extract invariants in terms of symbolic dynamics (Hubbard trees, spiders, Newton trees, etc) and classifies these systems in combinatorial terms; conversely, in order to show that such a symbolic dynamical system is actually realized in holomorphic dynamics, Thurston's theorem is the powerful tool of choice.
Very lively and productive interactions have developed recently between symbolic dynamics and group theory in the context of "iterated monodromy groups", a novel tool to investigate combinatorial properties of holomorphic dynamics that has had some impressive results; see the discussion.
Some of our main results:
- A classification of all kneading sequences that occur for complex quadratic polynomials, as well as with their Bernoulli measure:
Henk Bruin, Dierk Schleicher: Admissibility of kneading sequences and structure of Hubbard trees for quadratic polynomials. Journal of the London Mathematical Society 78 2 (2008), 502-522. (arxiv)
Henk Bruin, Dierk Schleicher: Bernoulli measure of complex admissible kneading sequences. Ergodic Theory and Dynamical Systems 33 3 (2013), 821-830. (arxiv)
- A classification of all postcritically finite rational maps that are Newton maps of polynomials:
Russell Lodge, Yauhen “Zhenya” Mikulich, and Dierk Schleicher: A classification of postcritically finite Newton maps. Submitted. (arxiv)
- The proof of Thurston's conjecture that core entropy of quadratic polynomials depends continuously on the parameter or the external angle (with an independent proof by Guilio Tiozzo):
Dzmitry Dudko, Dierk Schleicher: Core entropy of quadratic polynomials. (arxiv);
and of the related topic that all polynomials with connected Julia sets have biaccessibility dimension less than one, and hence core entropy less than the logarithm of the degree:
Philipp Meerkamp, Dierk Schleicher: Hausdorff dimension and biaccessibility for polynomial Julia sets. Proceedings of the American Mathematical Society 141 2 (2013), 533-542. (arxiv)
- A study of symbolic dynamics properties of the family of complex exponential maps:
Dierk Schleicher, Johannes Zimmer: Escaping points of exponential maps. Journal of the London Mathematical Society 67 2 (2003), 380-400. (article)
Markus Förster, Lasse Rempe, and Dierk Schleicher. Classification of escaping exponential maps. Proceedings of the American Mathematical Society 136 2 (2008), 651-663. (arxiv)
Bastian Laubner, Dierk Schleicher, and Vlad Vicol: A combinatorial classification of postsingularly finite complex exponential maps. Discrete and Continuous Dynamical Systems 22 3 (2008), 663-682. (arxiv)