28 research outputs found
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
The MacLane class and the Eremenko-Lyubich class
In 1970 G. R. MacLane asked if it is possible for a locally univalent function in the class A to have an arc tract. This question remains open, but several results about it have been given. We significantly strengthen these results, in particular replacing the condition of local univalence by the more general condition that the set of critical values is bounded. Also, we adapt a recent powerful technique of C. J. Bishop in order to show that there is a function in the Eremenko-Lyubich class for the disc that is not in the class A
Multiply connected wandering domains of entire functions
The dynamical behaviour of a transcendental entire function in any periodic
component of the Fatou set is well understood. Here we study the dynamical
behaviour of a transcendental entire function in any multiply connected
wandering domain of . By introducing a certain positive harmonic
function in , related to harmonic measure, we are able to give the first
detailed description of this dynamical behaviour. Using this new technique, we
show that, for sufficiently large , the image domains contain
large annuli, , and that the union of these annuli acts as an absorbing
set for the iterates of in . Moreover, behaves like a monomial
within each of these annuli and the orbits of points in settle in the long
term at particular `levels' within the annuli, determined by the function .
We also discuss the proximity of and for large
, and the connectivity properties of the components of . These properties are deduced from new results about the behaviour
of an entire function which omits certain values in an annulus
Dynamics of meromorphic functions with direct or logarithmic singularities
We show that if a meromorphic function has a direct singularity over
infinity, then the escaping set has an unbounded component and the intersection
of the escaping set with the Julia set contains continua. This intersection has
an unbounded component if and only if the function has no Baker wandering
domains. We also give estimates of the Hausdorff dimension and the upper box
dimension of the Julia set of a meromorphic function with a logarithmic
singularity over infinity. The above theorems are deduced from more general
results concerning functions which have "direct or logarithmic tracts", but
which need not be meromorphic in the plane. These results are obtained by using
a generalization of Wiman-Valiron theory. The method is also applied to complex
differential equations.Comment: 29 pages, 2 figures; v2: some overall revision, with comments and
references added; to appear in Proc. London Math. So
Baker's conjecture for functions with real zeros
Baker's conjecture states that a transcendental entire functions of order less than 1/2 has no unbounded Fatou components. It is known that, for such functions, there are no unbounded periodic Fatou components and so it remains to show that they can also have no unbounded wandering domains. Here we introduce completely new techniques to show that the conjecture holds in the case that the transcendental entire function is real with only real zeros, and we prove the much stronger result that such a function has no orbits consisting of unbounded wandering domains whenever the order is less than 1. This raises the question as to whether such wandering domains can exist for any transcendental entire function with order less than 1.
Key ingredients of our proofs are new results in classical complex analysis with wider applications. These new results concern: the winding properties of the images of certain curves proved using extremal length arguments, growth estimates for entire functions, and the distribution of the zeros of entire functions of order less than 1
Permutable entire functions and multiply connected wandering domains
Let f and g be permutable transcendental entire functions. We use a recent analysis of the dynamical behaviour in multiply connected wandering domains to make progress on the long standing conjecture that the Julia sets of f and g are equal; in particular, we show that J(f)=J(g) provided that neither f nor g has a simply connected wandering domain in the fast escaping set
The iterated minimum modulus and conjectures of Baker and Eremenko
In transcendental dynamics significant progress has been made by studying points whose iterates escape to infinity at least as fast as iterates of the maximum modulus. Here we take the novel approach of studying points whose iterates escape at least as fast as iterates of the minimum modulus, and obtain new results related to Eremenko's conjecture and Baker's conjecture, and the rate of escape in Baker domains. To do this we prove a result of wider interest concerning the existence of points that escape to infinity under the iteration of a positive continuous function