328 research outputs found
A landing theorem for entire functions with bounded post-singular sets
Get PDFThe Douady-Hubbard landing theorem for periodic external rays is one of the
cornerstones of the study of polynomial dynamics. It states that, for a complex
polynomial with bounded postcritical set, every periodic external ray lands at
a repelling or parabolic periodic point, and conversely every repelling or
parabolic point is the landing point of at least one periodic external ray.
We prove an analogue of this theorem for an entire function with bounded
postsingular set. If the function has finite order of growth, then it is known
that the escaping set contains certain curves called "periodic hairs"; we show
that every periodic hair lands at a repelling or parabolic periodic point, and
conversely every repelling or parabolic periodic point is the landing point of
at least one periodic hair. For a postsingularly bounded entire function of
infinite order, such hairs may not exist. Therefore we introduce certain
dynamically natural connected sets, called "filaments". We show that every
periodic filament lands at a repelling or parabolic periodic point, and
conversely every repelling or parabolic periodic point is the landing point of
at least one periodic filament.
More generally, we prove that every point of a hyperbolic set is the landing
point of a filament.Comment: 59 pages, 3 figures. Updated from the version in Geom. Funct.
Analysis with the term "filament," with an explanation of the change of
terminology. A small number of corrections has also been made from the
pre-publication manuscrip
Escaping sets are not sigma-compact
Get PDFLet be a transcendental entire function. The escaping set consists
of those points that tend to infinity under iteration of . We show that
is not -compact, resolving a question of Rippon from 2009.Comment: 6 pages. To appear in Proc. Amer. Math. Soc. V4: Author accepted
manuscript. Clarification of an imprecise statement regarding nowhere density
in Corollary 2.2 that was present in v
Singular orbits and Baker domains
Get PDFWe show that there is a transcendental meromorphic function with an invariant Baker domain such that every singular value of is a super-attracting periodic point. This answers a question of Bergweiler from 1993. We also show that can be chosen to contain arbitrarily large round annuli, centred at zero, of definite modulus. This answers a question of Mihaljevi\'c and the author from 2013, and complements recent work of Bara\'nski et al concerning this question
Singular orbits and Baker domains
Get PDFWe show that there is a transcendental meromorphic function with an invariant Baker domain U such that every singular value of f is a super-attracting periodic point. This answers a question of Bergweiler from 1993. We also show that U can be chosen to contain arbitrarily large round annuli, centred at zero, of definite modulus. This answers a question of Mihaljević and the author from 2013, and complements recent work of Barański et al concerning this question
Hereditarily indecomposable Julia continua of transcendental entire functions
Get PDFThis thesis studies the topology of the Julia set of transcendental entire functions of disjoint type. It is known that the Julia set of such entire functions may contain topological objects which could
be considered ``pathological''. In this sense, we may ask how pathological the Julia set could become.
Here, we prove the existence of a transcendental entire function of disjoint type for which the connected
components of its Julia set together with infinity are pseudo-arcs. Furthermore, the disjoint-type
entire function can be chosen to have finite lower order of growth
Singular orbits and Baker domains
Get PDFWe show that there is a transcendental meromorphic function with an invariant Baker domain such that every singular value of is a super-attracting periodic point. This answers a question of Bergweiler from 1993. We also show that can be chosen to contain arbitrarily large round annuli, centred at zero, of definite modulus. This answers a question of Mihaljević and the author from 2013, and complements recent work of Barański et al concerning this question.8 pages; to appear in Mathematische Annalen. V2: Minor revisions, changes and correction
The Eremenko-Lyubich constant
Get PDFEremenko and Lyubich proved that an entire function whose set of singular
values is bounded is expanding at points where its image has large modulus.
These expansion properties have been at the centre of the subsequent study of
this class of functions, now called the Eremenko-Lyubich class. We improve the
estimate of Eremenko and Lyubich, and show that the new estimate is
asymptotically optimal. As a corollary, we obtain an elementary proof that
functions in the Eremenko-Lyubich class have lower order at least
Points of convergence -- music meets mathematics
Get PDF"Phase-locking" is a fundamental phenomenon in which coupled or periodically
forced oscillators synchronise. The Arnold family of circle maps, which
describes a forced oscillator, is the simplest mathematical model of
phase-locking and has been studied intensively since its introduction in the
1960s. The family exhibits regions of parameter space where phase-locking
phenomena can be observed. A long-standing question asked whether "hyperbolic"
parameters~-- those whose behaviour is dominated by periodic attractors, and
which are therefore stable under perturbation~-- are dense within the family. A
positive answer was given in 2015 by van Strien and the author, which implies
that, no matter how chaotic a map within the family may behave, there are
always systems with stable behaviour nearby. This research was a focal point of
a pioneering collaboration with composer Emily Howard, commencing with Howard's
residency in Liverpool's mathematics department in 2015. The collaboration
generated impacts on creativity, culture and society, including several musical
works by Howard, and lasting influence on artistic practice through a
first-of-its-kind centre for science and music. We describe the research and
the collaboration, and reflect on the factors that contributed to the latter's
success.Comment: 6 pages, 3 figures. This is a preprint of the chapter "Points of
convergence - music meets mathematics," to appear in "More UK Success Stories
in Industrial Mathematics," ed. Philip J.\ Asto
Rational maps represented by both rabbit and aeroplane matings
No full textUnderstanding parameter spaces of rational maps is an active area of complex dynamics. There is a region of a particular parameter space of rational maps which contains all possible matings with the rabbit polynomial in a well understood manner. In an effort to further understand which hyperbolic components of the parameter space correspond to matings with the aeroplane we relate the family of matings with the aeroplane to the family of matings with the rabbit.
We present an algorithm, described in chapter 3, which calculates the mating with the rabbit which is Thurston equivalent to a given post-critically finite mating with the aeroplane. Chapter 4 gives a result describing which matings with the rabbit are Thurston equivalent to some mating with the aeroplane. Chapter 5 studies the algorithm in more detail, giving results bounding the number of steps required for the algorithm to produce a result
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Hausdorff dimensions of escaping sets of transcendental entire functions
No full textSuppose that f and g are transcendental entire functions, each with a bounded set of singular values, and that f and g are affinely equivalent. We show that the escaping sets of f and g have the same Hausdorff dimension.
Using a result of the second author, we deduce that there exists a family of transcendental entire functions for which the escaping set has Hausdorff dimension equal to one
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