We extend Thurston’s topological characterisation theorem for postcritically finite rational maps to a class of rational maps which have a fixed bounded type Siegel disk. This makes a small step towards generalizing Thurston’s theorem to geometrically infinite rational maps.

We show that there exist non-landing exponential rays with bounded accumulation sets. By introducing folding models of certain rays, we prove that each of the corresponding accumulation sets is an indecomposable continuum containing part of the ray, an indecomposable continuum disjoint from the ray or a Jordan arc.

We prove that for typical rotation numbers $0<{\it\theta}<1$, the boundary of the Siegel disk of $f_{{\it\theta}}(z)=e^{2{\it\pi}i{\it\theta}}\sin (z)$ centered at the origin is a Jordan curve which passes through exactly two critical points ${\it\pi}/2$ and $-{\it\pi}/2$.

We construct ergodic exponential maps fλ(z)=λez such that the forward orbit of the origin is dense in .

Let 0<θ<1 be an irrational number of bounded type. We prove that for any map in the family (e2πiθz+αz2)ez, α∈ℂ, the boundary of the Siegel disk, with fixed point at the origin, is a quasi-circle passing through one or both of the critical points.

Let F(z) be a rational map with degree at least three. Suppose that there exists an annulus such that (1) H separates two critical points of F, and (2) F:H→F(H) is a homeomorphism. Our goal in this paper is to show how to construct a rational map G by twisting F on H such that G has the same degree as F and, moreover, G has a Herman ring with any given Diophantine type rotation number.

Let fλ (z) = λez . In this short note, we consider those maps f λ with λ close to 1. We show that the probability that fλ is hyperbolic approaches 1 as λ → 1.