The dynamical fine structure of iterated cosine maps and a dimension paradox

摘要

We discuss in detail the dynamics of maps z bar right arrow ae(z) + be(-z) for which both critical orbits are strictly preperiodic. The points that converge to infinity under iteration contain a set R consisting of uncountably many curves called rays, each connecting infinity to a well-defined "landing point" in C, so that every point in C 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 C 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 R of rays has Hausdorff dimension 1, and each point in CR is connected to infinity by one or more disjoint rays in R. As the complement of a 1-dimensional set, CR of course has Hausdorff dimension 2 and full Lebesgue measure.