The dynamics of transcendental functions in the complex plane has received a significant amount of attention. In particular much is known about the description of Fatou components. Besides the types of periodic Fatou components that can occur for polynomials, there also exist so-called Baker domains, periodic components where all orbits converge to infinity, as well as wandering domains. In trying to find analogues of these one dimensional results, it is not clear which higher dimensional transcendental Communicated by Ngaiming Mok.

While the dynamics of transcendental entire functions in periodic Fatou components and in multiply connected wandering domains are well understood, the dynamics in simply connected wandering domains have so far eluded classification. We give a detailed classification of the dynamics in such wandering domains in terms of the hyperbolic distances between iterates and also in terms of the behaviour of orbits in relation to the boundaries of the wandering domains. In establishing these classifications, we obtain new results of wider interest concerning non-autonomous forward dynamical systems of holomorphic self maps of the unit disk. We also develop a new general technique for constructing examples of bounded, simply connected wandering domains with prescribed internal dynamics, and a criterion to ensure that the resulting boundaries are Jordan curves. Using this technique, based on approximation theory, we show that all of the nine possible types of simply connected wandering domain resulting from our classifications are indeed realizable.

We study the distribution of periodic points for a wide class of maps, namely entire transcendental functions of finite order and with bounded set of singular values, or compositions thereof. Fix p ∈ N and assume that all dynamic rays which are invariant under f p land. An interior p-periodic point is a fixed point of f p which is not the landing point of any periodic ray invariant under f p . Points belonging to attracting, Siegel or Cremer cycles are examples of interior periodic points. For functions as above, we show that rays which are invariant under f p , together with their landing points, separate the plane into finitely many regions, each containing exactly one interior p−periodic point or one parabolic immediate basin invariant under f p . This result generalizes the GoldbergMilnor Separation Theorem for polynomials [GM], and has several corollaries. It follows, for example, that two periodic Fatou components can always be separated by a pair of periodic rays landing together; that there cannot be Cremer points on the boundary of Siegel disks; that "hidden components" of a bounded Siegel disk have to be either wandering domains or preperiodic to the Siegel disk itself; or that there are only finitely many non-repelling cycles of any given period, regardless of the number of singular values.

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We consider higher-derivative perturbations of quantum gravity and quantum field theories in curved space and investigate tools to calculate counterterms and short-distance expansions of Feynman diagrams.In the case of single higher-derivative insertions we derive a closed formula that relates the perturbed one-loop counterterms to the unperturbed Schwinger-DeWitt coefficients. In the more general case, we classify the contributions to the short-distance expansion and outline a number of simplification methods.Certain difficulties of the common differential technique in the presence of higher-derivative perturbations are avoided by a systematic use of the Campbell-Baker-Hausdorff formula, which in some cases reduces the computational effort considerably.Expanding (1.1) around a background metric, the one-loop Feynman diagrams are encoded in the determinant of a differential operator containing higher-derivative terms. In general, the higher-derivative terms can be treated perturbatively or non-perturbatively. In the former approach [1, 2, 3] ("quantum gravity") they are viewed as perturbations of the Einstein lagrangian: the theory is non-renormalizable, but perturbatively unitary. In the latter approach [4, 5, 6] ("higher-derivative gravity") they are used to improve the behavior of Green functions at short distances: the theory is renormalizable, but not unitary. Here we are interested in the former approach, which is equivalent to study the insertions of higher-derivative operators in the Feynman diagrams of quantum gravity. Observe that in (1.1) higher powers of the curvature tensor generate perturbations with an arbitrary number of derivatives.