New Examples of Polynomial Julia Sets of Positive Area

Friday, October 31, 2014 - 11:00am - 12:00pm
Keller 3-180
Misha Lyubich (State University of New York, Stony Brook (SUNY))
The problem of Lebesgue area of polynomial Julia sets goes back to classical
work of Fatou. When the polynomial is reasonably hyperbolic then the Julia set
has zero area, and there are plenty of examples of this kind. First examples of
polynomial Julia sets of posiive area were constructed around 2006 by Buff and
Cheritat. However, these examples are rare and topologically wild.

In the talk, we will describe a new class of examples, certain
Feigenbaum Julia sets, that are tame and observable, with
explicit topological models and computable images. An extra curious
feature of these examples is that their hyperbolic dimension is strictly
less than the Hausdorff dimension. The corresponding parameter set has
a reasonable size (positive Hausdorff dimension).

On the other hand, we can construct examples of Feigenbaum Julia sets
whose Hausdorff dimension is strictly less than two.

Existence of such Julia sets goes against intuition coming from hyperbolic
geometry and theory of Kleinian groups.

It is a joint work with Artur Avila.
MSC Code: