Abstracts
INVITED TALKS
Adam EPSTEIN (Warwick University)
A Newhouse Phenomenon in Transcendental Dynamics
In joint work with Lasse Rempe-Gillen, we prove the existence of bounded type entire maps with infinitely many attractors. We obtain such bounded type maps by surgery on finite type maps with appropriate dynamics. We obtain additional control when the latter are selected from the $cos \sqrt{z}$ family.
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Dan NICKS (University of Nottingham)
Fast escape using normal families
Eremenko proved that for a transcendental entire function the escaping set is non-empty. In particular, he used Wiman-Valiron theory to construct points with forward orbits that tend to infinity fast, compared to an iterated maximum modulus. Research in "quasiregular dynamics" aims to generalise results from complex dynamics by studying the iteration of quasiregular mappings of R^n. Such quasiregular maps are a natural generalisation of entire functions, although no version of Wiman-Valiron theory is available. However, many results about normal families (including Montel's theorem) do generalise to this new setting and these have been used to construct fast escaping points for quasiregular maps. This approach gives an alternative method of proof in the original complex setting and, in fact, this leads us on to a result about the asymptotic rate of escape that appears to be new even for entire functions.
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Mary REES (University of Liverpool)
Persistent Markov Partitions in Complex Dynamics
Markov partitions are ubiquitous in dynamics, usually simply for providing recognisable dynamical systems up to conjugacy. But Markov partitions are especially useful in complex dynamics, because they can be used, not only to describe the dynamics of individual dynamical systems, but also to describe variation of dynamics on open subsets of parameter space. In fact what is used is not a single Markov partition but a sequence of them, which has come to be known as a puzzle, following Yoccoz, who first used what is known as the Yoccoz puzzle for quadratic polynomials. The corresponding sequence of puzzles of parameter space is known as the (Yoccoz) parapuzzle. I shall say something about the history of the use of such puzzles, and about a fairlly general construction of such puzzles which I have been studying.
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Gwyneth STALLARD (The Open University)
The structure of the escaping set of a transcendental entire function
The escaping set of a transcendental entire function consists of the set of points that escape to infinity under iteration and, in recent years, has come to play a key role in complex dynamics. Much work has been motivated by Eremenko's conjecture that all the components of the escaping set are unbounded. Significant progress has been made by studying the set of points that escape as fast as possible and it is known that all the components of the fast escaping set A_R(f) are unbounded. We show that, either A_R(f) has uncountably many components (as for the exponential function) or it is connected and has the form of a spiders web.
This is joint work with Phil Rippon.
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Sebastian VAN STRIEN (Imperial College London)
Quasi-symmetric rigidity in one-dimensional dynamics and applications
In this talk I will discuss quasisymmetric rigidity in real dimension one. Building on earlier work with Weixiao Shen
and Oleg Kozlovski, recently Trevor Clark and I have shown that under some additional very weak conditions two topologically conjugate C^3 interval maps are quasi symmetrically conjugate.
CONTRIBUTED TALKS
Simon ALBRECHT (Christian-Albrechts-Universität zu Kiel)
On the construction of entire functions in the Speiser class
It was proven by Gwyneth Stallard that for any 1<d<2 there exists a transcendental entire function with bounded set of singular values such that the Hausdorff dimension of its Julia set is exactly d. So far it is unknown if the same statement holds for transcendental entire functions with only finitely many singular values.
To approach this problem, we construct transcendental entire functions with only two singular values and a prescribed tract by using quasiconformal folding, a method introduced by C. Bishop in 2011.
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Matthieu ASTORG (Université Toulouse III Paul Sabatier)
An example of a polynomial endomorphism of C^2 with a wandering Fatou component
Joint work with X. Buff, R. Dujardin, H. Peters and J. Raissy.
In complex dimension one, the celebrated No Wandering Domain from Sullivan (1985) asserts that
every Fatou component is preperiodic. On the other hand, it is known that transcendental entire maps
may have wandering Fatou components. Building on ideas from Lyubich, we will explain how to construct
examples of polynomial mappings of C^2 that have a wandering Fatou component. Additionally, in these
examples, the wandering domains will be bounded.
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Markus BAUMGARTNER (Christian-Albrechts-Universität zu Kiel) *
On boundaries of multiply connected wandering domains
We show that under certain conditions the boundary of a multiply connected wandering domain of a transcendental entire function consists of a countable number of rectifiable Jordan curves.
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Fabrizio BIANCHI (Université Toulouse III Paul Sabatier and Università di Pisa)
Holomorphic motion for the Julia sets of holomorphic families of endomorphisms of CP(k)
We build measurable holomorphic motions for Julia sets of holomorphic families of endomorphisms of CP(k) under various equivalent notions of stability. This generalizes the well-known result obtained by Mane-Sad-Sullivan and Lyubich in dimension 1 and leads to a coherent definition of the bifurcation locus in this setting. Since the usual
1-dimensional techniques no longer apply in higher dimension, our approach is based on ergodic and pluripotential methods.
This is a joint work with François Berteloot.
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Alexandre DE ZOTTI (University of Liverpool)
Eventual hyperbolic dimension for transcendental functions
We recall some results on the rigidity of dynamics of transcendental entire functions in the class B near infinity, obtained by Lasse Rempe-Gillen and Gwyneth Stallard. Those results are based on quasiconformal rigidity of the dynamics of class B functions near infinity. Different dimension quantities are known to be invariant in affine equivalence classes of maps. Poincaré maps shows that this invariance may not hold for quasiconformal classes. This is a joint work with Lasse Rempe-Gillen.
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Matthew JACQUES (The Open University)
Semigroups of Möbius transformations
Motivated by the well-established theory of Kleinian groups and questions on the convergence of composition sequences, we explore the dynamics of semigroups of Möbius maps acting on the unit n-ball. We build upon theory first developed by Fried, Marotta and Stankewitz, and use it to study certain families of composition sequences. In dimension 2 we give necessary and sufficient conditions for convergence of these sequences.
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John OSBORNE (The Open University)
Connectedness properties of the set where the iterates of an entire function are unbounded
This talk will look at some connectedness properties of the set of points where the iterates of an entire function f are unbounded. In particular, we show that this set is connected if iterates of the minimum modulus of f tend to infinity.
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Remus RADU (Stony Brook University)
Complex Hénon maps with a semi-parabolic fixed point
We give a characterization of the family of complex Hénon maps with a semi-parabolic fixed point that arise as small perturbations of a quadratic polynomial p with a parabolic fixed point. We prove that the Hénon map has connected Julia set J which is homeomorphic to a quotiented solenoid; the Julia set J^+ fibers over the Julia set of the polynomial p. We will explain where these maps sit in the whole parameter space of complex Hénon maps and explore other regions of connectivity, with similar properties. This is joint work with Raluca Tanase.
Daniel SOMMERFELD (Christian-Albrechts-Universität zu Kiel)
Newton's method for certain entire functions
In this talk, we consider Newton's method for functions of type f(z)=p(z)exp(z)-1, where p is a polynomial of degree d. The goal is to specify a set of starting points for Newton's iteration, depending only on d and maximal modulus of the zeros of p. For this purpose, we give an approximation of any simple zero, whose modulus is large enough.
Moreover, we show that the immediate basins of attraction of these zeros are located in a horizontal strip of finite height. Furthermore, we give an idea how to construct the starting points for the zeros of small modulus.
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Raluca TANASE (Stony Brook University)
Continuity of Julia sets in C^2
We discuss some continuity results for the Julia sets J and J^+ of the complex Hénon map. We look at the parameter space of Hénon maps which have a fixed point with one eigenvalue (1+t)w, where w is a root of unity and t is real and sufficiently small. The Hénon map has a semi-parabolic fixed point when t is 0, and we use techniques that we have developed for the semi-parabolic case to describe nearby perturbations. We prove a two-dimensional analogue of radial convergence for polynomial Julia sets. This is joint work with Remus Radu.
Dimitra TSIGKARI (Université Pierre et Marie Curie - Paris 6)
Introduction to the dynamics of holomorphic endomorphisms of CP^k
This is an introductory talk on the dynamics in several complex variables, mostly for those who are not quite familiar with the subject. We are going to adopt a classical approach by using some tools of the pluripotential theory, such as currents. More precisely, we will define the Green current and its relation to the Julia set. Then, we will see some interesting dynamical results.
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Sebastian VOGEL (Christian-Albrechts-Universität zu Kiel)
On a quasiregular map with non-escaping set of finite measure
Generalizing a result of McMullen (1987), we show that the escaping set of a higher dimensional quasiregular analogue of sine introduced by Bergweiler and Eremenko (2011) has positive Lebesgue measure. Suitable composition with a quasiregular powermapping yields a quasiregular map with non-escaping set of finite measure.
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Mairi WALKER (The Open University)
Gaussian integer continued fractions, the Picard group, and hyperbolic geometry
It is well known that continued fractions can be viewed as compositions of Möbius transformations, and this leads to a representation of them using hyperbolic geometry. In this talk we will use this representation to study continued fractions with Gaussian integer coefficients, which relate firstly to elements of the Picard group, the group of Möbius transformations with Gaussian integer coefficients, and secondly to paths in a graph that arises naturally in hyperbolic geometry.
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* Markus Baumgartner was voted the best student talk of the conference, congratulations!
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