ICMS

The role of complex analysis in complex dynamics

May 20, 2013 - May 24, 2013

ICMS, 15 South College Street

Organisers

Name	Institution
Langley, James	University of Nottingham
Rippon, Philip	Open University
Stallard, Gwyneth	Open University

This workshop will enable progress on fundamental questions in complex dynamics by focusing on important links with complex analysis. Leading experts will present overviews of several deep techniques in complex analysis that have already had significant applications in complex dynamics, and discuss outstanding problems of major significance to which these techniques might be applied successfully.

Invited Participants (including Keynote Speakers) include:
Krzysztof Baranski,University of Warsaw
Alan Beardon, University of Cambridge
David Drasin, University of Purdue
Adam Epstein, University of Warwick
Alexandre Eremenko, University of Purdue
Núria Fagella, University of Barcelona
Alastair Fletcher, Northern Illinois University
Aimo Hinkkanen, University of Illinois
Xavier Jarque, University of Barcelona
Boguslawa Karpinska, Warsaw University of Technology
Janina Kotus, Warsaw University of Technology
Jim Langley, University of Nottingham
Dan Nicks, University of Nottingham
Lasse Rempe-Gillen, University of Liverpool
Phil Rippon, Open University
Dierk Schleicher, Jacobs University, Bremen
Mitsuhiro Shishikura, University of Kyoto
Gwyneth Stallard, Open University
Mariusz Urbanski, University of North Texas
Anna Zdunik, University of Warsaw
Jian-Hua Zheng, Tsinghua University

Arrangements

The workshop will commence with Registration 08.45- 09.30 on Monday and close on Friday afternoon at 15.30.

Participation
Applications for this workshop are now closed.   Please email  audrey.brown(at)icms.org.uk   if you have any enquiries.

Venue and Talks
The workshop will be held at 15 South College Street, Edinburgh.  You may also find this map useful for the workshop.  All lectures will be held in the Newhaven Lecture Theatre.    The Lecture Theatre is equipped with a data projector, computer, visualiser (the new generation of overhead projectors) and two blackboards.  The projector and one board may be used simultaneously.  It is best to bring your presentation on a memory stick to use in our ICMS computer.   Alternatively, it is possible for you to use your own laptop with our dataprojector, but please be aware that you may have to alter your laptop resolutions/settings. 

 

Travel
Information about travel to the UK and Edinburgh is available here.

Please note that it is your responsibility to have adequate travel insurance to cover medical and other emergencies (volcanic ash disruptions) that may occur on your trip.

A taxi directly from the airport will cost approximately 16.00 to 20.00 GBP to the city centre for a one-way journey.     There is also a bus service direct from the airport to the city centre which will cost 3.50 single or 6.00 GBP return - the Airlink 100.  This is a frequent service (every 10 minutes during peak times) and will bring you close to Waverley Railway Station, only a short walk to the accommodation and the workshop venue (see map in 'Venue' section above). 

Lothian buses charge £1.50 for a single, £3.50 for a day ticket. Please note that the exact fare is required and no change is given.

If travelling by train, please note that Edinburgh has two railway stations - Waverley Railway Station being the main station and closest to the workshop venue at 15 South College Street. If you alight at Edinburgh Waverley, the workshop venue is an easy 10 minute walk over North and South Bridge. The second railway station is called Haymarket and is at the West End of the city centre.

 

UK Visas
If you are travelling from overseas you may require an entry visa. A European visa does not guarantee entry to the UK. Please use this link to the UK Visas site to find out if you need a visa and if so how to apply for one.

 

Accommodation

• For Invited Participants (including Keynote Speakers): if requested, accommodation has been arranged by ICMS and details of your accommodation emailed to you.

• For Graduate Students and Postdocs: ICMS will have emailed details of your accommodation if you requested and were granted financial assistance in the form of accommodation.

• For all other participants: you should arrange and pay for your own accommodation (only once your application has been accepted).   Many participants will be staying at Pollock Halls of Residence so you may wish to consider this as your first option.   Some suggestions are below:

  Masson House, Pollock Halls of Residence, The University of Edinburgh, 18 Holyrood Park Road, Edinburgh EH16 5AY  (approx 70.00 GBP per night)
  0800 028 7118 (UK only) - +44 (0)131 651 2189 - http://www.book.accom.ed.ac.uk/
  University of Edinburgh accommodation. Pollock Halls is about a 20-25 minute walk.
  Masson House at Pollock Halls is very suitable and many of the participants will stay there.

  Jurys Inn Edinburgh, 43 Jeffrey Street, Edinburgh EH1 1DH (approx 75.00-90.00 GBP per night)
  +44 (0)131 200 3300 - www.jurys-edinburgh-hotels.com
  Only a 5 minute walk from ICMS.
  

  Edinburgh City Hotel, 79 Lauriston Place, Edinburgh, EH3 9HZ  (approx 70-80 GBP per night)
  +44 (0)131 622 7979 - http://www.edinburghcityhotel.com/
  

  St Christophers Hostel, 9-13 Market Street, Edinburgh EH1 1DE  (approx 25.00 or less depending on offers available)
  +44 (0)20 7407 1856 -  www.st-christophers.co.uk - bookings@st-christophers.co.uk
  This is Youth Hostel accommodation.
  

Further details and many more suggestions for accommodation can be found here 

 

Poster Session
The Poster Session will be held after the talks on Tuesday 21 May, at 18.00, in the Chapterhouse on the ground floor of ICMS.  At this time there will be an informal wine reception and time for all participants to view and discuss posters. Full details of the Poster Session were emailed (on 6 May) to all who expressed an interest in presenting a poster.

 

Catering
The workshop grant will cover refreshments throughout the event, lunch on Monday, Tuesday, Thursday and Friday (Wednesday being a 'free' afternoon), an informal wine reception with the Poster Session on Tuesday evening and a Workshop Dinner on Thursday evening.   The Workshop Dinner will be held in The Magnum Restaurant, 1 Albany Street, Edinburgh, EH1 3PY, at 19.30 on Thursday 23 May.  

 

Wireless Access
The workshop venue, 15 South College Street, has wireless access throughout. On arrival at Registration you will be given instructions and a code for accessing the wireless network.  For those without laptops, there will also be a couple of computers available for you to check your emails. 

 

Registration Fee
A 50.00 GBP registration fee is payable by all.  Unfortunately, we do not yet have an on-line payment system at ICMS.   Details of how to pay have been emailed to all on 6 May 2013.    

 

Programme

Please note that the first two talks of each day are survey talks on an area of complex analysis, followed by the applications of that area to complex dynamics.

 

Monday 20 May 2013

08.45-09.30	Registration & coffee
09.30-10.30	Alan Beardon (University of Cambridge ) The Hyperbolic Metric
10.30-11.00	Coffee/Tea in Chapterhouse, Level 1
11.00-12.00	Gwyneth Stallard (Open University) Applications of the hyperbolic metric in complex dynamics
12.00-14.00	Lunch (provided in the Chapterhouse) and discussion
14.00-14.45	Jian-Hua Zheng (Tsinghua University) Hyperbolic metric and complex dynamics
14.45-15.30	Anna Zdunik (University of Warsaw) Harmonic measure on non-homogenous Cantor sets
15.30-16.00	Coffee/Tea in Chapterhouse
16.00-16.45	Dan Nicks (University of Nottingham) The Julia set in quasiregular dynamics
16.45-17.30	Alastair Fletcher (Northern Illinois University) The fast escaping set and quasiregular mappings

 

Tuesday 21 May

09.30-10.30	James Langley (University of Nottingham ) An introduction to Nevanlinna theory
10.30-11.00	Coffee/Tea in Chapterhouse
11.00-12.00	Aimo Hinkkanen (University of Illinois at Urbana-Champaign) Applications of Nevanlinna theory in complex dynamics
12.00-14.00	Lunch (provided in the Chapterhouse) and discussion
14.00-14.45	Krzysztof Barański (University of Warsaw) Absorbing domains for meromorphic maps
14.45-15.30	Boguslawa Karpińska (Warsaw University of Technology) Connectivity of Julia sets of meromorphic maps
15.30-16.00	Coffee/Tea in Chapterhouse
16.00-16.20	Anand Prakash Singh (Central University of Rajasthan) Spiraling Baker domain
16.20-16.40	David Martí Pete (Universitat de Barcelona) The escaping set of holomorphic self-maps of C*
16.40-17.00	Sebastian Vogel (University of Kiel) On the measure of the escaping set of a quasiregular analogue of the sine function
17.00-17.20	John Osborne (Open University) Connectedness properties of the set K(f) where the iterates of an entire function f are bounded
17.20-17.50	John Mayer (University of Alabama at Birmingham) The Central Strip Lemma for laminations of degree d>2: identity return polygons
18.00-19.30	Poster Session and Wine Reception

 

Wednesday 22 May

09.30-10.30	Alexandre Eremenko (University of Purdue) An introduction to inverse function singularities
10.30-11.00	Coffee/Tea in Chapterhouse
11.00-12.00	Núria Fagella (University of Barcelona) Introduction to quasiconformal surgery
12.00-13.00	Mitsuhiro Shishikura (Kyoto University) Introduction to quasiconformal surgery

 

Thursday 23 May

09.30-10.30	Lasse Rempe-Gillen (University of Liverpool) Constructing polynomials and entire functions by quasiconformal folding
10.30-11.00	Coffee/Tea in Chapterhouse
11.00-12.00	Xavier Jarque (University of Barcelona) Bishop's quasiconformal folding and wandering domains in Eremenko-Lyubich class
12.00-14.00	Lunch (provided in the Chapterhouse) and discussion
14.00-14.45	Adam Epstein (University of Warwick) Dynamics and deformations of finite type analytic maps
14.45-15.30	Dierk Schleicher (Jacobs University, Bremen) TBA
15.30-16.00	Coffee/Tea in Chapterhouse
16.00-16.20	Simon Albrecht (University of Kiel) On the construction of entire functions in the Speiser class
16.20-16.40	Dave Sixsmith (Open University) The size of the Julia set of some families of functions outside the Eremenko-Lyubich class
16.40-17.00	Markus Baumgartner (University of Kiel) On boundaries of multiply connected wandering domains
17.00-17.30	Tuen Wai Ng (University of Hong Kong) On finite Blaschke products sharing preimage of connected compact sets
19.30	Dinner at The Magnum Restaurant, 1 Albany Street, Edinburgh, EH1 3PY

 

Friday 24 May

09.30-10.30	David Drasin (Purdue University) Introduction and background to harmonic and subharmonic functions
10.30-11.00	Coffee/Tea in Chapterhouse
11.00-12.00	Philip Rippon (Open University) Applications of harmonic functions in complex dynamics
12.00-13.00	Lunch (provided in the Chapterhouse)
13.00-13.30	Shaun Bullett (Queen Mary, University of London) The discreteness locus for a family of correspondences
13.30-14.15	Janina Kotus (Warsaw University of Technology) The Hausdorff dimemsion of the set of escaping paramaters and the escaping set
14.15-15.00	Mariusz Urbanski (University of North Texas) Regularity and irregularity of fiber dimensions of non-autonomous holomorphic dynamical systems
15.00-15.30	Coffee/Tea in Chapterhouse

 

Presentations:

Presentation Details
Albrecht, Simon
On the construction of entire functions in the Speiser class
View Abstract Hide Abstract
We construct entire functions with finite set of singular values and prescribed tracts by using quasiconformal folding, a method introduced by C. Bishop in 2012.
Badeńska, Agnieszka
POSTER: Functions with real multipliers of all repelling periodic orbits
Barański, Krzysztof
Absorbing domains for meromorphic maps
View Abstract Hide Abstract
Extending a result of Cowen for holomorphic self-maps of the disc, we show the existence of absorbing domains for holomorphic self-maps of hyperbolic regions whose iterates tend to a boundary point. The result is used in the proof of the fact that every transcendental meromorphic map with a Baker domain and a disconnected Julia set has a weakly repelling fixed point. This is a joint work with Nuria Fagella, Xavier Jarque and Boguslawa Karpinska.
Baumgartner, Markus
On boundaries of multiply connected wandering domains
View Abstract Hide Abstract
We show that the boundary of a multiply connected wandering domain of a certain family of transcendental entire functions is a countable number of Jordan curves. This family was first considered by Bergweiler and Zheng in 2011 and includes the first example of a multiply connected wandering domain by Baker.
Beardon, Alan
The hyperbolic metric
View Abstract Hide Abstract
I shall give an intuitive description of negative curvature, and some of the important features of hyperbolic geometry, and accompany these with some useful formulae with applications but no proofs.
Bullett, Shaun
The discreteness locus for a family of correspondences
View Abstract Hide Abstract
The discreteness locus for a family of holomorphic correspondences is a subset of parameter space analogous to a Bers slice for a family of Kleinian groups. We investigate the structure of this locus for a one-parameter family of correspondences containing matings and exhibit examples of striking dynamical behaviour at boundary points.
Canela, Jordi
POSTER: Tongues in degree 4 Blaschke products
Drasin, David
An introduction and background to harmonic and subharmonic functions
View Abstract Hide Abstract
This is an expository lecture to prepare non-experts for some of the talks on the iteration of entire functions. The talk begins recalling the definition of harmonic and subharmonic functions, the Dirichlet problem, harmonic measure. We will also discuss the extended maximum principle (polar sets) and if time permits, the Milloux-Schmidt inequality which provides one form of a precise version of the maximum principle.
Epstein, Adam
Dynamics and deformations of finite type analytic maps
Eremenko, Alexandre
An introduction to inverse function singularities
View Abstract Hide Abstract
An elementary introduction to the singularities of the inverse functions will be given, together with a survey of classical and modern results. The theory of these singularities has rich interaction with holomorphic dynamics which will be addressed in other talks. This talk can be considered a background to the talk of W. Bergweiler.
Fagella, Núria
Introduction to quasiconformal surgery
View Abstract Hide Abstract
We give the basic background on quasiconformal mappings and quasiconformal surgery, as a technique to construct maps with prescribed dynamics. We also give some examples of surgery constructions which will later be complemented by those in the talk that follows, by Mitsuhiro Shishikura.
Fletcher, Alastair
The fast escaping set and quasiregular mappings
View Abstract Hide Abstract
The fast escaping set for transcendental entire functions is a subject of ongoing interest in complex dynamics. In the spirit of exploring dynamical notions which make sense in higher real dimensions, in this talk we will discuss the fast escaping set for quasiregular mappings of transcendental type, and give examples where the fast escaping set has the structure of a higher dimensional spider's web. (This is joint work with David Drasin and Walter Bergweiler).
Hinkkanen, Aimo
Applications of Nevanlinna theory in complex dynamics
View Abstract Hide Abstract
We will survey results in complex dynamics whose proofs involve applications of the Nevanlinna theory of value distribution. Topics include the density of repelling fixed points in the Julia set, repelling fixed points of iterates, permutable entire functions, factorization of functions, and the Hausdorff dimension of the Julia sets of certain entire functions.
Jarque, Xavier
Bishop's quasiconformal folding and wandering domains in Eremenko-Lyubich class
View Abstract Hide Abstract
In the first part of the talk we will give an overview about the existence and non existence of wandering domains for different families of maps. In the second part of the talk we will explain Bishop's construction of a wandering (oscillating) domain for a map in Eremenko-Lyubich class.
Karpińska, Boguslawa
Connectivity of Julia sets of meromorphic maps
View Abstract Hide Abstract
In this talk we show that every transcendental meromorphic map with a multiply connected Baker domain has a weakly repelling fixed point. This was the last remaining case in the proof of the fact that every transcendental meromorphic map with a disconnected Julia set has a weakly repelling fixed point. These results imply that the Julia set of Newton's method for finding zeroes of an entire map is connected. This is a joint work with Krzysztof Baranski, Nuria Fagella and Xavier Jarque.
Kisaka, Masashi
POSTER: Transcendental entire functions whose Julia sets are Sierpinski carpets
Kotus, Janina
The Hausdorff dimemsion of the set of escaping paramaters and the escaping set
View Abstract Hide Abstract
Let $f$ be a non-constant elliptic function. We consider the escaping parameters in the family $f_b = b f$, i.e. the parameters  for which the orbit of one critical value of f escapes to infinity. Under additional assumptions on f we prove that the Hausdorff dimension of the set of escaping parameters in the family f is equal to the Hausdorff dimension of the escaping set in the dynamical space. The result also holds for one-parameter families of simply periodic meromorphic functions.
Langley, James
An introduction to Nevanlinna theory
View Abstract Hide Abstract
The talk will be an introduction to Nevanlinna theory for meromorphic functions of one complex variable and some of the key results which have played a significant role in complex dynamics.
Martí Pete, David
The escaping set of holomorphic self-maps of C*
View Abstract Hide Abstract
We are interested in the iteration of holomorphic self-maps of the punctured plane C*=C{0} for which both zero and infinity are essential singularities. These maps are of the form f(z)=z^n*exp(g(z)+h(1/z)) with n entire and g(z), h(z) nonconstant entire functions. In particular, we would like to understand what is the structure of the escaping set I(f), the set of points whose orbit accumulates to zero and/or infinity. In the setting of transcendental entire functions, A. Eremenko conjectured that every escaping point could be joined with infinity by a curve in I(f). In our case, we see that for functions of bounded type satisfying certain technical conditions the escaping points can be joined either to zero or to infinity by dynamic rays. Finally, we relate this property to the notion of order of growth of the function.
Mayer, John
The Central Strip Lemma for laminations of degree d>2: identity return polygons
View Abstract Hide Abstract
Quadratic laminations of the unit disk were introduced by Thurston as a vehicle for understanding the (connected) Julia sets of quadratic polynomials and the parameter space of quadratic polynomials. The "Central Strip Lemma" plays a key role in Thurston's classification of gaps in quadratic laminations, and in describing the corresponding parameter space. We generalize the notion of Central Strip to laminations of degree d>2 and prove a Central Strip Lemma for degree d>2. We conclude with an application of the Central Strip Lemma to higher degree laminations, in particular to identity return polygons, that shows it may play a role similar to Thurston's Central Strip Lemma in understanding higher degree laminations.
Mukherjee, Sabyasachi
POSTER: Towards a combinatorial description of the multicorns
Ng, Tuen Wai
On finite Blaschke products sharing preimage of connected compact sets
View Abstract Hide Abstract
In this talk, we shall consider the problem of characterizing those finite Blaschke products sharing preimage of connected compact subsets with positive hyperbolic capacity of the standard unit disk. We shall see that this problem is closely related to the factorization of finite Blaschke products (in the sense of compositions).
Nicks, Dan
The Julia set in quasiregular dynamics
View Abstract Hide Abstract
Quasiregular maps of R^n are a natural generalization of analytic functions on the complex plane. Hence one may desire an iterative theory for quasiregular maps analogous to the familiar results of complex dynamics. However, the usual definition of the Julia set as the region of non-normality does not, in general, transfer well to the quasiregular setting. Instead we may base a definition on the 'blowing-up' property of Julia sets of complex functions, i.e. the forward orbit of any neighbourhood of a Julia point is nearly the whole space.
Osborne, John
Connectedness properties of the set K(f) where the iterates of an entire function f are bounded
View Abstract Hide Abstract
We describe a class of transcendental entire functions for which K(f) is totally disconnected if and only if each component of K(f) containing a critical point is aperiodic. For the same class of functions, we also show that if K(f) is disconnected then it has uncountably many components.
Penrose, Christopher
POSTER: Super-statistics of Blaschke products
View Abstract Hide Abstract
Suppose we are given two $C^2$ orientation-preserving circle maps $A$ and $B$ which both expand Lebesgue measure $lambda$ on the unit circle. Given an arbitrary composition $C = C_lcircldotscirc C_1$ (or word) with $C_in{A,B}$ we obtain the unique invariant probability measure $lambda_C$ equivalent to $lambda$. Considering the special case of iteration alternating between blocks $C_i=A$ of length $m$ and $C_i=B$ of length $n$ (averaged over ending phases) we obtain the Birkhoff measure $$ frac{1}{m+n}{sum_{i=1}^mlambda_{A^icirc B^ncirc A^{m-i}} + sum_{j=1}^nlambda_{B^jcirc A^mcirc B^{n-j}}} . $$ We are interested in the statement that as $m$ and $n$ tend to infinity with fixed ratio $p:q$ (satisfying $p+q=1$) this Birkhoff measure tends to the super-statistical limit $plambda_A + qlambda_B$. We examine this convergence in the special case of $A$ and $B$ degree two Blaschke products.
Pérez-Garrandés, Carlos
POSTER: Ergodicity of laminations in compact surfaces
View Abstract Hide Abstract
A lamination by Riemann surfaces is a set which can be locally seen as a product of a complex disk by a topological set. This local structure induces a partition by Riemann surfaces, called leaves. We are interested in ergodic properties of these sets. It is very unusual to find closed currents associated with these objects, but in an article published in 2002, Bo Berndtsson and Nessim Sibony proved that laminations by Riemann surfaces with hyperbolic singularities allow positive harmonic currents. Afterwards, John Erik Fornaess and Nessim Sibony developed a intersection theory for harmonic currents which lead them to prove the uniqueness of these currents for laminations with hyperbolic singularities embedded in the complex projective plane under some generic assumptions. We will present how to use this theory in order to prove the generalization of this result for every compact Kähler surface.
Rempe-Gillen, Lasse
Constructing polynomials and entire functions by quasiconformal folding
View Abstract Hide Abstract
Bishop recently developed an extremely powerful method for constructing entire function with good control of the set of singular values. In particular, it is now possible to construct functions with finitely many (in fact, two) singular values with essentially any imaginable function-theoretical behaviour. I shall try and give a brief overview of the relevant results and the idea of the proof. To explain the latter, I shall focus on Bishop’s original application: given an arbitrary compact and connected subset K of the plane, construct a polynomial with only two singular values whose set of critical points is close to K in the Hausdorff metric. Given time, I shall also mention some recent applications of this method: the construction of entire functions whose Julia set is a bouquet of pseudo-arcs, and the construction of "Belyi functions" (branched coverings of the sphere that are branched over only three points) on arbitrary non-compact surfaces.
Rippon, Philip
Applications of harmonic functions in complex dynamics
View Abstract Hide Abstract
In this talk we discuss three applications of harmonic and subharmonic functions to the dynamical behaviour of transcendental entire functions. Firstly, we use harmonic functions to describe the dynamical behaviour of iterates of points in a multiply connected Fatou component. Secondly, we indicate how they can be used to obtain a regularity condition on the maximum modulus of certain transcendental entire functions, which has implications for their dynamical behaviour. Finally, we use results about harmonic measure to obtain results about the existence of escaping points on the boundaries of escaping wandering domains.
Shishikura, Mitsuhiro
Introduction to quasiconformal surgery
View Abstract Hide Abstract
Following Nuria Fagella's talk, we discuss further examples of quasiconformal surgery, which include pasting Sigels disks to make Herman rings of various configurations, Branner-Douady surgery. We also discuss the problem of conformal moduli and the continuity/discontinuity of the surgery.
Singh, Anand Prakash
Spiraling Baker domain
View Abstract Hide Abstract
Let $f$ be a transcendental entire function. For $ n in mathbb{N}$, let $f^n$ denote the $n^{th} $ iterate of $f$. Fatou set $F(f)$ of $f$ is defined to be the set of all points $z$ in the complex plane $mathbb{C}$ such that the family $ {f^n}_{n geq 1} $ forms a normal family in some neighbourhood of $z$. Julia set is defined to be the complement of Fatou set. A periodic component $U$ of $F(f)$ of period $m$ is called a Baker domain if $ f^{mn}(z) rightarrow infty $ as $n rightarrow infty $ for all $z in U$. Further we define a Baker domain $B$ as a positively oriented spiraling Baker domain if there exist positive continuous functions $ A(r), phi (r), psi(r) $, of $r$ all tending to $ infty$ as $ r rightarrow infty $ such that $phi (r) $ in non decreasing, $ 0 < psi (r) - phi (r) < 2 pi $ and $ B subset G := {textrm{ } A(r) e^{i theta}textrm{ } |textrm{ } phi(r) < theta < psi(r),textrm{ } r > r_1}$ for some $r_1 > 0.$ Similarly we define a negatively oriented Baker domain. By Spiraling Baker domain, we mean either positively oriented Spiraling Baker domain or a negatively oriented spiraling Baker domain. In this paper we give some interesting properties of Spiraling Baker domain.
Sixsmith, Dave
The size of the Julia set of some families of functions outside the Eremenko-Lyubich class
View Abstract Hide Abstract
We briefly discuss some new results regarding the size of the Julia and escaping sets of some families of functions which are not in the Eremenko-Lyubich class. For some of these functions the Julia set is a spider's web.
Stallard, Gwyneth
Applications of the hyperbolic metric in complex dynamics
View Abstract Hide Abstract
In this talk we discuss three applications of the hyperbolic metric in complex dynamics. Firstly, we show how it is used to prove a basic distortion property of Fatou components in the escaping set and discuss how this provides information on the rate of escape. Secondly, we discuss how the hyperbolic metric can be used to relate the geometry of a Fatou component to the rate of escape. Finally, we show how the hyperbolic metric can be used to prove two new covering results which can in turn be used to prove the existence of points in the Julia set with prescribed rates of escape.
Urbanski, Mariusz
Regularity and irregularity of fiber dimensions of non-autonomous holomorphic dynamical systems
View Abstract Hide Abstract
This talk will concern non-autonomous dynamics of rational functions and, more precisely, the fractal behavior of the Julia sets under perturbation of non-autonomous systems. We provide A necessary and sufficient condition for holomorphic stability will be provided and discussed. It leads to H"older continuity of dimensions of hyperbolic non-autonomous Julia sets with respect to the $l^infty$-topology on the parameter space. On the other hand, in drastic contradisctinction to the deterministic case, for some particular non-autonomous family of rational (functions, the Hausdorff and packing dimension functions are not differentiable at any point and these dimensions do not coincide on an open dense set of the parameter space still with respect to the $l^infty$-topology.
Vogel, Sebastian
On the measure of the escaping set of a quasiregular analogue of the sine function
View Abstract Hide Abstract
Generalizing a result of McMullen (1987), we show that the escaping set of a higher dimensional quasiregular analogue of the sine function introduced by Bergweiler and Eremenko (2011) has positive Lebesgue measure.
Zdunik, Anna
Harmonic measure on non-homogenous Cantor sets
View Abstract Hide Abstract
I will present the results on metric properties of the harmonic measure on certain Cantor sets, defined by non-autonomous dynamics. Some results on the Hausdorff measure on these sets will be also given. This is a joint work with Athanasios Batakis (Université d'Orleans).
Zheng, Jian-Hua
Hyperbolic metric and complex dynamics
View Abstract Hide Abstract
An important characteristic of hyperbolic domain is to have the hyperbolic metric over it, hence no holomorphic functions map non-hyperbolic domain into a hyperbolic domain (Picard theorem is a special case) and a holomorphic function is "expanding" with respect to the hyperbolic metric if it is not covering, otherwise, it is "isometric". From the point of view of dynamics, it is natural that the hyperbolic metric become a powerful tool in complex dynamics. In fact, more and more results have been obtained in terms of the hyperbolic metric. In this talk, I begin with some inequalities from the hyperbolic metric and then discuss their applications in complex dynamics.

Participants

Name	Institution
Albrecht, Simon	University of Kiel
Badeńska, Agnieszka	Warsaw University of Technology
Barański, Krzysztof	University of Warsaw
Baumgartner, Markus	University of Kiel
Beardon, Alan	University of Cambridge
Blokh, Alexander	University of Alabama at Birmingham
Bullett, Shaun	Queen Mary, University of London
Canela, Jordi	University of Barcelona
Drasin, David	Purdue University
Epstein, Adam	University of Warwick
Eremenko, Alexandre	University of Purdue
Fagella, Núria	University of Barcelona
Fletcher, Alastair	Northern Illinois University
Hinkkanen, Aimo	University of Illinois at Urbana-Champaign
Jacques, Matthew	Open University
Jarque, Xavier	University of Barcelona
Karpińska, Boguslawa	Warsaw University of Technology
Kisaka, Masashi	Kyoto University
Kotus, Janina	Warsaw University of Technology
Langley, James	University of Nottingham
Lodge, Russell	Jacobs University Bremen
Mamayusupov, Khudoyor	Jacobs University Bremen
Martí Pete, David	University of Barcelona
Mayer, John	University of Alabama at Birmingham
Morosawa, Shunsuke	Kochi University
Mukherjee, Sabyasachi	Jacobs University Bremen
Nada, Alhabib	University of Liverpool
Ng, Tuen Wai	University of Hong Kong
Nicks, Dan	University of Nottingham
Osborne, John	Open University
Oversteegen, Lex	University of Alabama at Birmingham
Penrose, Christopher	Queen Mary, University of London
Pérez-Garrandés, Carlos	Universidad Complutense de Madrid
Rempe-Gillen, Lasse	University of Liverpool
Rippon, Philip	Open University
Shishikura, Mitsuhiro	Kyoto University
Short, Ian	Open University
Singh, Anand Prakash	Central University of Rajasthan
Sixsmith, Dave	Open University
Sommerfeld, Daniel	Christian-Albrechts-Universität zu Kiel
Stallard, Gwyneth	Open University
Subedi, Bishnu Hari	Tribhuvan University, Kathmandu
Urbanski, Mariusz	University of North Texas
Vogel, Sebastian	University of Kiel
Zdunik, Anna	University of Warsaw
Zheng, Jian-Hua	Tsinghua University