295 research outputs found
Boundaries of univalent Baker domains
Let be a transcendental entire function and let be a univalent Baker domain of . We prove a new result about the boundary behaviour of conformal maps and use this to show that the non-escaping boundary points of form a set of harmonic measure zero with respect to . This leads to a new sufficient condition for the escaping set of to be connected, and also a new general result on Eremenko's conjecture
Connectedness properties of the set where the iterates of an entire function are unbounded
We investigate the connectedness properties of the set I+(f) of points where the iterates of an entire function f are unbounded. In particular, we show that I+(f) is connected whenever iterates of the minimum modulus of f tend to ∞. For a general transcendental entire function f, we show that I+(f)∪ \{\infty\} is always connected and that, if I+(f) is disconnected, then it has uncountably many components, infinitely many of which are unbounded
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
Slow escaping points of quasiregular mappings
This article concerns the iteration of quasiregular mappings on Rd and entire functions on C. It is shown that there are always points at which the iterates of a quasiregular map tend to infinity at a controlled rate. Moreover, an asymptotic rate of escape result is proved that is new even for transcendental entire functions. Let f:Rd→Rd be quasiregular of transcendental type. Using novel methods of proof, we generalise results of Rippon and Stallard in complex dynamics to show that the Julia set of f contains points at which the iterates fn tend to infinity arbitrarily slowly. We also prove that, for any large R, there is a point x with modulus approximately R such that the growth of |fn(x)| is asymptotic to the iterated maximum modulus Mn(R,f)
Entire functions with Julia sets of positive measure
Let f be a transcendental entire function for which the set of critical and
asymptotic values is bounded. The Denjoy-Carleman-Ahlfors theorem implies that
if the set of all z for which |f(z)|>R has N components for some R>0, then the
order of f is at least N/2. More precisely, we have log log M(r,f) > (N/2) log
r - O(1), where M(r,f) denotes the maximum modulus of f. We show that if f does
not grow much faster than this, then the escaping set and the Julia set of f
have positive Lebesgue measure. However, as soon as the order of f exceeds N/2,
this need not be true. The proof requires a sharpened form of an estimate of
Tsuji related to the Denjoy-Carleman-Ahlfors theorem.Comment: 17 page
Escape rate and Hausdorff measure for entire functions
The escaping set of an entire function is the set of points that tend to
infinity under iteration. We consider subsets of the escaping set defined in
terms of escape rates and obtain upper and lower bounds for the Hausdorff
measure of these sets with respect to certain gauge functions.Comment: 24 pages; some errors corrected, proof of Theorem 2 shortene
Rigidity of escaping dynamics for transcendental entire functions
We prove an analog of Boettcher's theorem for transcendental entire functions
in the Eremenko-Lyubich class B. More precisely, let f and g be entire
functions with bounded sets of singular values and suppose that f and g belong
to the same parameter space (i.e., are *quasiconformally equivalent* in the
sense of Eremenko and Lyubich). Then f and g are conjugate when restricted to
the set of points which remain in some sufficiently small neighborhood of
infinity under iteration. Furthermore, this conjugacy extends to a
quasiconformal self-map of the plane.
We also prove that this conjugacy is essentially unique. In particular, we
show that an Eremenko-Lyubich class function f has no invariant line fields on
its escaping set.
Finally, we show that any two hyperbolic Eremenko-Lyubich class functions f
and g which belong to the same parameter space are conjugate on their sets of
escaping points.Comment: 28 pages; 2 figures. Final version (October 2008). Various
modificiations were made, including the introduction of Proposition 3.6,
which was not formally stated previously, and the inclusion of a new figure.
No major changes otherwis
Are Devaney hairs fast escaping?
Beginning with Devaney, several authors have studied transcendental entire
functions for which every point in the escaping set can be connected to
infinity by a curve in the escaping set. Such curves are often called Devaney
hairs. We show that, in many cases, every point in such a curve, apart from
possibly a finite endpoint of the curve, belongs to the fast escaping set. We
also give an example of a Devaney hair which lies in a logarithmic tract of a
transcendental entire function and contains no fast escaping points.Comment: 22 pages, 1 figur
Historic landscape character and sense of place
This is an Author's Accepted Manuscript of an article published in Landscape Research, 2013, Vol. 38, Issue 2 pp.179-202, copyright Taylor & Francis, available online at: http://www.tandfonline.com/10.1080/01426397.2012.672642.Most studies of landscape character within archaeology and historical geography have focused on morphological features such as whether settlement patterns were nucleated or dispersed, but this paper discusses how adding depth to this, for example by studying place-names, vernacular architecture, and the territorial structures within which a landscape was managed in the past, gives us a far greater understanding of its texture and meaning to local communities. In two case-studies in southern Essex, for example, it is shown how the connections that once existed between inland and coastal communities can be used today to promote public access to the countryside. A further case study, in southwest England, shows how field-/place-names and vernacular architecture also make an important contribution to our appreciation of the time depth and complexity of landscape character.Arts and Humanities Research Council (AHRC)Southend-on-Sea Borough Counci
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