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The resultant on compact Riemann surfaces
We introduce a notion of resultant of two meromorphic functions on a compact
Riemann surface and demonstrate its usefulness in several respects. For
example, we exhibit several integral formulas for the resultant, relate it to
potential theory and give explicit formulas for the algebraic dependence
between two meromorphic functions on a compact Riemann surface. As a particular
application, the exponential transform of a quadrature domain in the complex
plane is expressed in terms of the resultant of two meromorphic functions on
the Schottky double of the domain.Comment: 44 page
Hausdorff dimension of escaping sets of meromorphic functions
We give a complete description of the possible Hausdorff dimensions of
escaping sets for meromorphic functions with a finite number of singular
values. More precisely, for any given we show that there exists
such a meromorphic function for which the Hausdorff dimension of the escaping
set is equal to . The main ingredient is to glue together suitable
meromorphic functions by using quasiconformal mappings. Moreover, we show that
there are uncountably many quasiconformally equivalent meromorphic functions
for which the escaping sets have different Hausdorff dimensions.Comment: 37 pages, 8 figures. Some overall revision in the introduction. More
details added in Section
Iteration of meromorphic functions
This paper attempts to describe some of the results obtained in the iteration
theory of transcendental meromorphic functions, not excluding the case of
entire functions. The reader is not expected to be familiar with the iteration
theory of rational functions. On the other hand, some aspects where the
transcendental case is analogous to the rational case are treated rather
briefly here. For example, we introduce the different types of components of
the Fatou set that occur in the iteration of rational functions but omit a
detailed description of these types. Instead, we concentrate on the types of
components that are special to transcendental functions (Baker domains and
wandering domains).Comment: 38 pages. Abstract added in migration. See
http://analysis.math.uni-kiel.de/bergweiler/ for recent comments and
correction
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