Duke Math. J. 136 (2), 343-356, (01 February 2007) DOI: 10.1215/S0012-7094-07-13625-1
KEYWORDS: 37F35, 30D05, 37B10, 37C45, 37D45, 37F10, 37F20
We discuss in detail the dynamics of maps for which both critical orbits are strictly preperiodic. The points that converge to under iteration contain a set consisting of uncountably many curves called rays, each connecting to a well-defined “landing point” in , so that every point in is either on a unique ray or the landing point of several rays.
The key features of this article are the following:
(1) this is the first example of a transcendental dynamical system, where the Julia set is all of and the dynamics is described in detail for every point using symbolic dynamics;
(2) we get the strongest possible version (in the plane) of the “dimension paradox”: the set of rays has Hausdorff dimension , and each point in is connected to by one or more disjoint rays in .
As the complement of a -dimensional set, of course has Hausdorff dimension and full Lebesgue measure