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

Abstract

We discuss in detail the dynamics of maps $z\mapsto {\mathrm{ae}}^z+{\mathrm{be}}^{-z}$ for which both critical orbits are strictly preperiodic. The points that converge to $\infty$ under iteration contain a set $R$ consisting of uncountably many curves called rays, each connecting $\infty$ to a well-defined “landing point” in $\mathbb{C}$, so that every point in $\mathbb{C}$ 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 $\mathbb{C}$ 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 $\mathbb{C}\backslash R$ is connected to $\infty$ by one or more disjoint rays in $R$.

As the complement of a $1$-dimensional set, $\mathbb{C}\backslash R$ of course has Hausdorff dimension $2$ and full Lebesgue measure