We give an example of a dominant rational self-map of the projective plane whose dynamical degree is a transcendental number.

We discuss applications of an improvement on the Riemann mapping theorem which replaces the unit disc by another “double quadrature domain,” i.e., a domain that is a quadrature domain with respect to both area and boundary arc length measure. Unlike the classic Riemann mapping theorem, the improved theorem allows the original domain to be finitely connected, and if the original domain has nice boundary, the biholomorphic map can be taken to be close to the identity, and consequently, the double quadrature domain is close to the original domain. We explore some of the parallels between this new theorem and the classic theorem, and some of the similarities between the unit disc and the double quadrature domains that arise here. The new results shed light on the complexity of many of the objects of potential theory in multiply connected domains.

We construct Coleff–Herrera products and Bochner–Martinelli type residue currents associated with a tuple$f$of weakly holomorphic functions, and show that these currents satisfy basic properties from the (strongly) holomorphic case. This include the transformation law, the Poincaré–Lelong formula and the equivalence of the Coleff–Herrera product and the Bochner–Martinelli type residue current associated with$f$when$f$defines a complete intersection.

We prove a certain non-linear version of the Levi extension theorem for meromorphic functions. This means that the meromorphic function in question is supposed to be extendable along a sequence of complex curves, which are arbitrary, not necessarily straight lines. Moreover, these curves are not supposed to belong to any finite-dimensional analytic family. The conclusion of our theorem is that nevertheless the function in question meromorphically extends along an (infinite-dimensional) analytic family of complex curves and its domain of existence is a pinched domain filled in by this analytic family.

We extend a recent result of Avelin, Hed, and Persson about approximation of functions f that are plurisubharmonic on a domain Ω and continuous on ˙Ω, with functions that are plurisubharmonic on (shrinking) neighborhoods of ˙Ω. We show that such approximation is possible if the boundary of Ω is C0 outside a countable exceptional set E⊂∂Ω. In particular, approximation is possible on the Hartogs triangle. For Hölder continuous u, approximation is possible under less restrictive conditions on E. We next give examples of domains where this kind of approximation is not possible, even when approximation in the Hölder continuous case is possible.