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
Dierk Schleicher. "The dynamical fine structure of iterated cosine maps and a dimension paradox." Duke Math. J. 136 (2) 343 - 356, 01 February 2007. https://doi.org/10.1215/S0012-7094-07-13625-1