Walter Bergweiler
On the packing dimension of the Julia set and the escaping set of an entire function.
Wed, 01/19/2011 - 16:01 — SomNambulAuthors: Walter Bergweiler
Subjects: Complex Variables
AbstractWe show that for large classes of entire functions the Julia set and the
escaping set have packing dimension two. For example, this is the case for
entire functions which are bounded on a curve tending to infinity. More
generally, we show that the result holds under suitable growth conditions
involving the minimum and maximum modulus.Dynamics of a higher dimensional analog of the trigonometric functions.
Tue, 02/23/2010 - 16:01 — SomNambulAuthors: Walter Bergweiler, Alexandre Eremenko
Subjects: Dynamical Systems
AbstractWe introduce a higher dimensional quasiregular map analogous to the
trigonometric functions and we use the dynamics of this map to define, for d>1,
a partition of d-dimensional Euclidean space into curves tending to infinity
such that two curves may intersect only in their endpoints and such that the
union of the curves without their endpoints has Hausdorff dimension one.Normal families and fixed points of iterates.
Tue, 01/26/2010 - 16:01 — SomNambulAuthors: Walter Bergweiler
Subjects: Complex Variables
AbstractLet F be a family of holomorphic functions and let K be a constant less than
4. Suppose that for all f in F the second iterate of f does not have fixed
points for which the modulus of the multiplier is greater than K. We show that
then F is normal. This is deduced from a result about the multipliers of
iterated polynomials.On the Hausdorff dimension of the Julia set of a regularly growing entire function.
Wed, 09/23/2009 - 13:19 — SomNambulAuthors: Walter Bergweiler, Bogusława Karpińska
Subjects: Dynamical Systems
AbstractWe show that if the growth of a transcendental entire function f is
sufficiently regular, then the Julia set and the escaping set of f have
Hausdorff dimension 2.On the Hausdorff dimension of the Julia set of a regularly growing entire function.
Wed, 09/23/2009 - 13:19 — SomNambulAuthors: Walter Bergweiler, Bogusława Karpińska
Subjects: Dynamical Systems
AbstractWe show that if the growth of a transcendental entire function f is
sufficiently regular, then the Julia set and the escaping set of f have
Hausdorff dimension 2.On the number of solutions of a transcendental equation arising in the theory of gravitational lensing.
Tue, 09/01/2009 - 12:47 — SomNambulAuthors: Walter Bergweiler, Alexandre Eremenko
Subjects: Complex Variables
AbstractThe equation in the title describes the number of bright images of a point
source under lensing by an elliptic object with isothermal density. We prove
that this equation has at most 6 solutions. Any number of solutions from 1 to 6
can actually occur.
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