151 research outputs found
Absorbing sets and Baker domains for holomorphic maps
We consider holomorphic maps for a hyperbolic domain in the
complex plane, such that the iterates of converge to a boundary point
of . By a previous result of the authors, for such maps there exist
nice absorbing domains . In this paper we show that can be
chosen to be simply connected, if has parabolic I type in the sense of the
Baker--Pommerenke--Cowen classification of its lift by a universal covering
(and is not an isolated boundary point of ). Moreover, we provide
counterexamples for other types of the map and give an exact
characterization of parabolic I type in terms of the dynamical behaviour of
Absorbing sets and Baker domains for holomorphic maps
We consider holomorphic maps for a hyperbolic domain in the complex plane, such that the iterates of converge to a boundary point of . By a previous result of the authors, for such maps there exist nice absorbing domains . In this paper we show that can be chosen to be simply connected, if has doubly parabolic type in the sense of the Baker-Pommerenke-Cowen classification of its lift by a universal covering (and is not an isolated boundary point of ). We also provide counterexamples for other types of the map and give an exact characterization of doubly parabolic type in terms of the dynamical behaviour of
On the connectivity of the Julia sets of meromorphic functions
We prove that every transcendental meromorphic map f with a disconnected
Julia set has a weakly repelling fixed point. This implies that the Julia set
of Newton's method for finding zeroes of an entire map is connected. Moreover,
extending a result of Cowen for holomorphic self-maps of the disc, we show the
existence of absorbing domains for holomorphic self-maps of hyperbolic regions
whose iterates tend to a boundary point. In particular, the results imply that
periodic Baker domains of Newton's method for entire maps are simply connected,
which solves a well-known open question.Comment: 34 pages, 10 figure
Virtual Immediate Basins of Newton Maps and Asymptotic Values
Newton's root finding method applied to a (transcendental) entire function
f:C->C is the iteration of a meromorphic function N. It is well known that if
for some starting value z, Newton's method converges to a point x in C, then f
has a root at x. We show that in many cases, if an orbit converges to infinity
for Newton's method, then f has a `virtual root' at infinity. More precisely,
we show that if N has an invariant Baker domain that satisfies some mild
assumptions, then 0 is an asymptotic value for f.
Conversely, we show that if f has an asymptotic value of logarithmic type at
0, then the singularity over 0 is contained in an invariant Baker domain of N,
which we call a virtual immediate basin. We show by way of counterexamples that
this is not true for more general types of singularities.Comment: 15 pages, 1 figur
Exotic Baker and wandering domains for Ahlfors islands maps
Let X be a Riemann surface of genus at most 1, i.e. X is the Riemann sphere
or a torus. We construct a variety of examples of analytic functions g:W->X,
where W is an arbitrary subdomain of X, that satisfy Epstein's "Ahlfors islands
condition". In particular, we show that the accumulation set of any curve
tending to the boundary of W can be realized as the omega-limit set of a Baker
domain of such a function. As a corollary of our construction, we show that
there are entire functions with Baker domains in which the iterates converge to
infinity arbitrarily slowly. We also construct Ahlfors islands maps with
wandering domains and logarithmic singularities, as well as examples where X is
a compact hyperbolic surface.Comment: 18 page
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