67 research outputs found
The escaping set of the exponential
We show that the points that converge to infinity under iteration of the
exponential map form a connected subset of the complex plane.Comment: 5 pages, 1 figur
An answer to a question of Herman, Baker and Rippon concerning Siegel disks
We give a positive answer to a question of Herman, Baker and Rippon by
showing that, in the exponential family, the boundary of every unbounded Siegel
disk contains the singular value.Comment: 4 pages, 1 figure; appeared as preprint 02/7 in the Berichtsreihe des
Mathematischen Seminars der CAU Kiel (2002
Topological Dynamics of Exponential Maps on their Escaping Sets
We develop an abstract model for the dynamics of an exponential map on its set of escaping points and, as an analog of Boettcher's
theorem for polynomials, show that every exponential map is conjugate, on a
suitable subset of its set of escaping points, to a restriction of this model
dynamics. Furthermore, we show that any two attracting and parabolic
exponential maps are conjugate on their sets of escaping points; in fact, we
construct an analog of Douady's "pinched disk model" for the Julia sets of
these maps. On the other hand, we show that two exponential maps are generally
not conjugate on their sets of escaping sets. Using the correspondence with our
model, we also answer several questions about escaping endpoints of external
rays, such as when a ray is differentiable in such an endpoints or how slowly
these endpoints can escape to infinity.Comment: 38 pages, 3 figures. // V3: Several typos fixed; some overall
revision; parts of the material in Sections 5, 7 and 11 have been rewritte
On Nonlanding Dynamic Rays of Exponential Maps
We consider the case of an exponential map for which the singular value is
accessible from the set of escaping points. We show that there are dynamic rays
of which do not land. In particular, there is no analog of Douady's ``pinched
disk model'' for exponential maps whose singular value belongs to the Julia
set.
We also prove that the boundary of a Siegel disk for which the singular
value is accessible both from the set of escaping points and from contains
uncountably many indecomposable continua.Comment: 15 pages; 1 figure. V2: A result on Siegel disks, as well as a
figure, has been added. Some minor corrections were also mad
Connected escaping sets of exponential maps
We show that for many complex parameters a, the set of points that converge
to infinity under iteration of the exponential map f(z)=e^z+a is connected.
This includes all parameters for which the singular value escapes to infinity
under iteration.Comment: 9 pages; minor revisions from Version
On a question of Eremenko concerning escaping components of entire functions
Let f be an entire function with a bounded set of singular values, and
suppose furthermore that the postsingular set of f is bounded. We show that
every component of the escaping set I(f) is unbounded. This provides a partial
answer to a question of Eremenko.Comment: 7 pages; 1 figure. V2: Final version (some minor changes
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
- …