Polynomial Julia sets with tree structure, typically Hubbard trees, play an important role in holomorphic dynamics. In this paper, we study the dynamics of ${f_\alpha (z)=z^2+\alpha \bar {z}}$ for $\alpha $ being real and the Julia sets being trees. We show that all such $\alpha $ form an interval $[1,4]$. This answers a question of G. Sienra. We further show that $f_\alpha $ exhibits non-trivial dynamics on the Fatou set which equals to the escaping set.

We prove a number of results concerning the Hausdorff and packing dimension of sets of points which escape (at least in average) to infinity at a given rate under non-autonomous iteration of exponential maps. In particular, we generalize the results proved by Sixsmith in 2016 and answer his question on annular itineraries for exponential maps.

We show that for any quasimeromorphic mapping with an essential singularity at infinity, there exist points whose iterates tend to infinity arbitrarily slowly. This extends a result by Nicks for quasiregular mappings, and Rippon and Stallard for transcendental meromorphic functions on the complex plane. We further establish a new result for the growth rate of quasiregular mappings near an essential singularity, and briefly extend some results regarding the bounded orbit set and the bungee set to the quasimeromorphic setting.

Let be a non-constant elliptic function. We prove that the Hausdorff dimension of the escaping set of f equals 2q/(q+1), where q is the maximal multiplicity of poles of f. We also consider the escaping parameters in the family fβ = βf, i.e. the parameters β for which the orbit of one critical value of fβ escapes to infinity. Under additional assumptions on f we prove that the Hausdorff dimension of the set of escaping parameters ε in the family fβ is greater than or equal to the Hausdorff dimension of the escaping set in the dynamical space. This demonstrates an analogy between the dynamical plane and the parameter plane in the class of transcendental meromorphic functions.

Let $f$ be a transcendental meromorphic function with at least one direct tract. In this note, we investigate the structure of the escaping set which is in the same direct tract. We also give a theorem about the slow escaping set.