Janina Kotus

Let where R is a non-constant rational map. We prove that if is not an asymptotic value of f, then the Hausdorff dimension of the escaping set of f equals , where q is the maximal multiplicity of poles of f. We also consider the set of escaping parameters in the family , i.e. the parameters for which the orbit of a pre-pole of f escapes to infinity under iterates of . Under additional assumptions on f we prove that the Hausdorff dimension of the set of escaping parameters in the family is greater than or equal to .

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 f be a transcendental meromorphic function of finite order ρ \rho for which the set of finite singularities of f − 1 f^{-1} is bounded. Suppose that ∞ \infty is not an asymptotic value and that there exists M ∈ N M \in \mathbb N such that the multiplicity of all poles, except possibly finitely many, is at most M M . For R > 0 R>0 let I R ( f ) I_R(f) be the set of all z ∈ C z\in \mathbb {C} for which lim inf n → ∞ | f n ( z ) | ≥ R \liminf _{n\to \infty }|f^n(z)|\geq R as n → ∞ n\to \infty . Here f n f^n denotes the n n -th iterate of f f . Let I ( f ) I(f) be the set of all z ∈ C z\in \mathbb {C} such that | f n ( z ) | → ∞ |f^n(z)|\to \infty as n → ∞ n\to \infty ; that is, I ( f ) = ⋂ R > 0 I R ( f ) I(f)=\bigcap _{R>0} I_R(f) . Denote the Hausdorff dimension of a set A ⊂ C A\subset \mathbb {C} by H D ( A ) \mathrm {HD}(A) . It is shown that lim R → ∞ H D ( I R ( f ) ) ≤ 2 M ρ / ( 2 + M ρ ) \lim _{R \to \infty } \mathrm {HD}(I_R(f))\leq 2 M \rho /(2+ M\rho ) ...

We construct examples of elliptic functions, viewed as iterated meromorphic functions from the complex plane to the sphere, with the property that there exist one or more critical points which approach the essential singularity at ∞ under iteration but are not prepoles. We obtain many nonequivalent elliptic functions satisfying this property, including examples with Julia set the whole sphere as well as examples with nonempty Fatou set. These are the first examples of this type known to exist and provide examples to illustrate unusual chaotic measure theoretic behaviour studied by the third author and others.

We study the existence of finite absolutely continuous invariant measures for meromorphic Misiurewicz maps whose Julia set is the whole sphere. In the rational context, these hypotheses imply that such a measure must exist. We show that it is not so for meromorphic maps unless an additional condition on the behavior of the map, which can be stated in terms of its Nevanlinna characteristic, is satisfied.

After the pioneering work on complex dynamics by Fatou and Julia in the early 20th century, Noel Baker went on to lay the foundations of transcendental complex dynamics. As one of the leading exponents of transcendental dynamics, he showed how developments in complex analysis such as Nevanlinna theory could be applied. His work has inspired many others to take up this increasingly active subject, and will continue to do so. Presenting papers by researchers in transcendental dynamics and complex analysis, this book is written in honour of Noel Baker. The papers describe the state of the art in this subject, with new results on completely invariant domains, wandering domains, the exponential parameter space, and normal families. The inclusion of comprehensive survey articles on dimensions of Julia sets, buried components of Julia sets, Baker domains, Fatou components of functions of small growth, and ergodic theory of transcendental meromorphic functions means this is essential reading for students and researchers in complex dynamics and complex analysis.

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