Programme

We welcome speakers who are early career researchers. If you are an early career researcher who wishes to give a talk then please get in touch with us. 

Upcoming seminars


Speaker: Dave Sixsmith, Open University

Title: TBA

Abstract: TBA


Past seminars


Speaker: Jamie Walton, University of Nottingham

Title: Self-similarity of cut and project sets

Abstract: Aperiodic Order studies the long-range structure of infinite patterns that lack periodicity but still exhibit structural order. Amongst other applications, aperiodically ordered tilings and point sets are used as models for physical quasicrystals and the field benefits from a rich interaction of various viewpoints, such as the study of these objects through dynamical systems. There are relatively few known general methods for constructing classes of them, the two main ones being the cut and project method, and tiling (or point set) substitution, which generates patterns that are ‘hierarchical’ or ‘self-similar’. It is of interest to know how these classes of patterns are related, and in particular which patterns can be constructed via both techniques; many famous examples fall into this category, such as the Penrose tilings and the recently discovered hat tilings. Knowing when a substitution tiling can be expressed using a (‘nice’) cut and project scheme is of relevance to the Pisot Conjecture. In this talk I will explain recent work, joint with Edmund Harriss and Henna Koivusalo, on the reverse direction: when is a cut and project set substitutive? As well as general Euclidean cut and project schemes we also consider the restricted setting of schemes with polytopal windows, obtaining a very simple characterisation that generalises some previous results in this area to general dimensions.


Speaker: Veronica Beltrami, University of Parma

Title: SCV: Transcendental Hénon maps with distinct and hyperbolic limit sets 

Abstract: The theory of holomorphic dynamics in one complex variable is much more advanced than what is known in several complex variables (SCV), where the situation is vastly different. Even in two-dimensional complex dynamics, the study of automorphisms already presents difficult challenges, and the construction of significant examples is an active area of research. The first part of the seminar lays the groundwork by introducing fundamental concepts of holomorphic dynamics in SCV, setting the stage for the second, more technical part. We will focus on the stable dynamics of non-polynomial automorphisms of C^2, specifically transcendental Hénon maps. We will be interested in analyzing escaping Fatou components with rank 1 limit functions and addressing two questions that were open until a few months ago:


Speaker: Gustavo Rodrigues Ferreira, Centre de Recerca Matematica Barcelona 

Title: Classifying multiply connected wandering domains 

Abstract: We develop a framework to unify the "hyperbolic-distortion-based" approach to internal dynamics of simply connected wandering domains (developed by Benini et al) and the mostly case-by-case approach to multiply connected wandering domains (previously developed by the first author). We do this by looking at the sequence of injectivity radii along an orbit in the domain, introducing geometric information about the shape of the wandering domains that interacts with the function-based information given by the hyperbolic distortion. This both unifies previous results and completes the description of the internal dynamics of any wandering domain of a meromorphic function. As a consequence, we conclude that the internal dynamics (from a hyperbolic-geometric point of view) of a wandering domain are of one of six kinds, and that five of these kinds can be realised by wandering domains of entire functions.


Speaker: Liz Vivas, Ohio State University 

Title: Dynamics of rational inner skew-products 

Abstract: I will define what are rational inner skew-products on two dimensions and then talk about the dynamics of a specific class of rational inner skew-products. This is joint work with with R. Birkett, UIC; J. Raissy, Bordeaux; and A. Sola, Stockholm.


Easter break


Speaker: Kasun Fernando, Brunel University London

Title: The Bootstrap for Dynamical Systems

Abstract: A dynamical system is usually represented by a probabilistic model of which the unknown parameters must be estimated using statistical methods. When measuring the uncertainty of such parameter estimations, the bootstrap stands out as a simple but powerful technique. In this talk, I will discuss the bootstrap for the Birkhoff averages of expanding maps and establish not only its consistency but also its second-order accuracy using the continuous first-order Edgeworth expansion.


Speaker: Matthieu Astorg, Université d'Orléans 

Title: Local dynamics of skew-products tangent to the identity (joint work with Luka Boc Thaler)

Abstract: The results we will present in this talk deal with local dynamics of skew-products P with a (non-degenerate) tangent to the identity fixed point at the origin. We will give an explicit sufficient condition on its coefficients for P to have wandering Fatou components. In particular, we will see that the dynamics of quadratic maps of the form (z,w)-> (z-z^2,w+w^2+bz^2) is surprisingly rich: under an explicit arithmetic condition on b, these maps have an infinity of grand orbits of wandering Fatou components, all of which admit non-constant limit maps. The main technical result is a parabolic implosion-type theorem, in which the renormalization limits that appear are different from previously known cases.


Speaker: Andrew Brown, University of Liverpool

Title: Constructing slow-growing counterexamples to the Strong Eremenko Conjecture 

Abstract: The strong Eremenko Conjecture was disproved in the paper of Rottenfußer, Rückert, Rempe and Schleicher with a counterexample function in the Eremenko—Lyubich Class B with infinite order growth. It was also shown in the same paper that the conjecture holds for functions of finite order growth (and finite compositions of them). This talk will discuss the construction of counterexample functions with infinite order growth that grow asymptotically to finite order functions.


Speaker: Gandhar Joshi, Open University

Title:  Monochromatic Arithmetic Progressions in the Fibonacci word

Abstract:  We study monochromatic arithmetic progressions (MAPs) in automatic sequences. This was initially inspired by Van der Waerden’s celebrated theorem in Ramsay theory. Nagai et al. (2021) proved that MAPs in a particular class of constant-length substitution fixed points are never infinite. We study the MAPs in non-constant length substitutions and prove that MAPs in the Fibonacci word are never infinite, as well as a few interesting numerical results using 'Walnut'.


Speaker: Leticia Pardo Simón, University of Manchester

Title: On the Hausdorff dimension of boundaries of attracting basins of entire maps

Abstract: Given an entire map f with an attracting cycle, the points that converge under iteration to such cycle form an open set, known as basin of attraction. The boundaries of such basins lie in the Julia set- the locus of chaos- of the map. In this talk we will discuss under which conditions on f such boundaries have Hausdorff dimension strictly larger than one. This is based on work in progress with K. Barański, B. Karpińska, D. Martí-Pete and A.  Zdunik.


Speaker: Julia Slipantschuk, University of Warwick

Title: Resonances for analytic expanding and hyperbolic maps

Abstract: Eigenvalues of transfer operators, known as Pollicott-Ruelle resonances provide insight into the long-term behaviour of the underlying dynamical system, in particular determining its exponential mixing rates. I will present a complete description of Pollicott-Ruelle resonances for a class of rational Anosov diffeomorphisms on the two-torus, inspired by the results in the one-dimensional expanding setting. This allows us to show that every homotopy class of two-dimensional Anosov diffeomorphisms contains (non-linear) maps with the sequence of resonances decaying stretched exponentially, exponentially or having only trivial resonances.


Christmas break


Speaker: Nikolai Prochorov, Aix-Marseille Université

Title: Towards Transcendental Thurston Theory 

Abstract: In the 1980’s Fields medallist, William Thurston obtained his celebrated characterization of post-critically finite rational maps. This result laid the foundation of such a field as Thurston's theory in holomorphic dynamics, which has been actively developing in the last few decades. One of the most important problems in this area is the characterization question, which asks whether a given topological map is equivalent to a holomorphic one. The result of W. Thurston and further developments allow us to answer this question quite effectively in the setting of (postcritically finite) maps of finite degree, and it has numerous applications for the dynamics of rational maps.

 

A similar question can be formulated for the maps of infinite degrees (i.e., in the transcendental setting), for instance, for entire postsingularly finite maps. However, the characterization problem becomes significantly more complicated, and the complete answer in the transcendental case is still not known.

 

In my talk, I am going to motivate the questions above and introduce the key notions of Thurston's theory in the transcendental setting. Further, I am going to explain the main techniques to attack the characterization problem in the transcendental case and provide some examples. Finally, I am going to report about new classes of maps, where we managed to obtain an answer to this question.


Speaker: Reem Yassawi, Queen Mary University of London

Title: Almost automorphic and bijective factors of substitution shifts

Abstract: Let f:(X,T)—>(Y,T') be a factor map of topological dynamical systems. We say that  (X,T) is an  almost automorphic extension if for some y in Y, the f-preimage of y is a singleton.  In the case where  (X,T) is  a finite-to-one extension of  (Y,T'), we say that this extension is  isometric   if f is k-to-1 for some k. Finally, a point x in X is distal if  there is no point whose T-orbit comes arbitrarily close to that of x.

Veech’s theorem tells us that any system with a residual set of distal points has an almost automorphic extension  which can be realised as an inverse limit of alternating isometric and almost automorphic extensions of the trivial (one point) system. We investigate this result for the special family of constant length substitution shifts. Our approach is algebraic: we define a  

finite semigroup S defined by the substitution. We characterise the existence of almost automorphic factors in terms of Green’s R-relation of S, and the existence of factors, which can lead to isometric extensions, in terms of Green’s L-relation of S. Our results are constructive. This is joint work with Álvaro Bustos-Gajardo and Johannes Kellendonk. 

I promise my talk will be elementary!


Speaker: Polina Vytnova, University of Surrey

Title: Computing the Hausdorff dimension of the Apollonian gasket

Abstract: We combine the ideas developed in a recent joint work with M. Pollicott on Hausdorff dimension estimates, (Transactions of the AMS, Ser. B, Vol. 9, pp. 1102--1159) with the approach to systems with neutral fixed points by C. Wormell (Efficient computation of statistical properties of intermittent dynamics, arXiv: 2106.01498) to rigorously compute the Hausdorff dimension of the Apollonian gasket.


Speaker: Ziyu (Nero) Li, Imperial College London

Title: Fractal dimensions for random substitution graph systems 

Abstract: This work aims to introduce fractal geometry into graph theory. To do this, we present substitution graph systems

and define box-counting dimension, Hausdorff dimension and especially degree dimension for graphs.

 

With the definitions, we prove that random substitution graph systems almost surely satisfy fractality and scale-freeness. Moreover, in deterministic cases, the associated box-counting and degree dimensions are analytically derived. For random ones, we obtain box-counting and degree dimensions numerically by the Lyapunov exponents.

In particular, Hausdorff and box-counting dimensions are proven to be consistent regarding substitution graph systems.

 

The random substitution graph systems turn out to be a simple, powerful and promising model either to theoretically study fractal graphs or to generate random fractal-like and scale-free networks.


Speaker: Roberto Florido Llinas, Universitat de Barcelona

Title:  An atlas of wandering domains for a family of transcendental Newton maps 

Abstract: Meromorphic maps naturally arise from Newton’s root-finding method applied to an entire function F. In the transcendental case, Newton’s method may particularly fail to converge to the roots of F if the initial condition lies in a Baker or wandering domain.
 In this talk, we present the simplest one-parameter family of transcendental entire functions with zeros, whose Newton’s method yields wandering domains for an open set of parameters by means of the logarithmic lifting method for periodic Fatou components. This is joint work with N. Fagella. 


Speaker: Bernhard Reinke, University of Liverpool

Title: Transcendental Wandering Triangles 

Abstract: A celebrated theorem of William Thurston states that every branch point of a locally connected Julia set of a quadratic polynomial is precritical or preperiodic.

In fact, the theorem is usually phrased in the language of quadratic laminations as the “No Wandering Triangle Theorem”.

Laminations are a tool in polynomial and nowadays also transcendental entire dynamics that capture the combinatorics of the landing behaviour of dynamic rays (or filaments), a wandering

triangle corresponds to three dynamic rays landing together at a point that is neither precritical nor preperiodic.

By the work of Blokh-Oversteegen and Buff-Canela-Roesch, it is known that there are cubic polynomials with wandering triangles.

 

In this talk, I will give an overview of joint work-in-progress with Jordi Canela and Lasse Rempe of constructing wandering triangles for transcendental entire functions, in particular in the family $a \cos \sqrt z + b$.


The Grimm Network

Holomorphic Dynamics Scheme 3 Meeting 


The role of wandering domains in Holomorphic Dynamics


Speaker: Maryam Hosseini, Queen Mary University of London

Title: Topological Factoring of Zero Dimensional Dynamical Systems


Abstract: Kakutani-Rokhlin partitions are appropriate tools in studying zero dimensional dynamical systems as these systems may appear in realizations of measure preserving ergodic systems on probability spaces. So the problem of existence of Kakutani-Rokhlin towers for a zero dimensional systems have been studied by people in Dynamical systems as well as operator algebraists which leads to creation of Dimension Group. In this talk, I will talk about application of Kakutani Rokhlin towers and dimension group in studying topological factoring between two zero dimensional systems. The talk is mostly based on a joint current work with Nasser Golestani.


Speaker: Becca Winarski, College of the Holy Cross

Title:  Thurston theory: unifying dynamical and topological

Abstract: Abstract: Thurston proved that a non-Lattés branched cover of the sphere to itself is either equivalent to a rational map (that is: conjugate via a mapping class), or has a topological obstruction. The Nielsen–Thurston classification of mapping classes is an analogous theorem in low-dimensional topology. We unify these two theorems with a single proof, further connecting techniques from surface topology and complex dynamics. Moreover, our proof gives a new framework for classifying self-covering spaces of the torus and Lattés maps. This is joint work with Jim Belk and Dan Margalit.


Speaker: Ronnie Pavlov, University of Denver

Title: Subshifts of very low complexity (slides)

 

Abstract: The word complexity function p(n) of a subshift X measures the number of n-letter words appearing in sequences in X, and X is said to have linear complexity if p(n)/n is bounded. It's been known since work of Ferenczi that subshifts X with linear word complexity function (i.e. limsup p(n)/n finite) have highly constrained/structured behavior. I'll discuss recent work with Darren Creutz, where we show that if limsup p(n)/n < 4/3, then the subshift X must in fact have measurably discrete spectrum, i.e. it is isomorphic to a compact abelian group rotation. Our proof uses a substitutive/S-adic decomposition for such shifts, and I'll touch on connections to the so-called S-adic Pisot conjecture.


Speaker: Jordi Canela, Universitat Jaume I 

Title: Achievable connectivities of Fatou components in singular perturbations 

 

Abstract: In this talk we will consider the dynamical system given by the iteration of a rational map Q over the Riemann Sphere. The dynamics of Q split the Riemann Sphere into two totally invariant sets. The Fatou set consists of all points z such that the family of iterates of Q is normal, or equivalently equicontinuous, in some open neighbourhood of z. The Fatou set is open and corresponds to the set of points with stable dynamics. Its complement, the Julia set, is closed and corresponds to the set of points which present chaotic behaviour. 

 Fatou components, connected components of the Fatou set, are mapped amongst themselves under iteration of Q. A periodic Fatou component can only have connectivity 1, 2, or infinity. Despite that, preperiodic Fatou components can have arbitrarily large finite connectivity. There exist explicit examples of rational maps with Fatou components of any prescribed connectivity. However, the degree of these maps grows as the required connectivity increases. 

We study a family of singular perturbations of rational maps with a single free critical point. Under certain conditions, the dynamical planes of these singular perturbations contain Fatou components of arbitrarily large finite connectivity. In this talk we will analyze the dynamical conditions under which these Fatou components of arbitrarily large connectivity appear.


Speaker: Rohini Ramadas, University of Warwick

Title: Degenerations and irreducibility problems in rational dynamics 

Abstract: Per_n is a (nodal) Riemann surface parametrizing degree-2 rational functions with an n-periodic critical point. The n-th Gleason polynomial G_n is a polynomial in one variable with Z-coefficients, whose roots correspond to degree-2 polynomials with an n-periodic critical point (i.e. to the period-n components of the Mandelbrot set). Two long-standing open questions are: (1) Is Per_n connected? (2) Is G_n is irreducible over Q? We show that if G_n is irreducible over Q, then Per_n is connected. In order to do this, we find a smooth point with Q-coordinates on a compactification of Per_n. This smooth Q-point represents a special degeneration of degree-2 rational maps, and as such admits an interpretation in terms of tropical geometry. 


Speaker: Argyrios Christodoulou, Aristotle University of Thessaloniki

Title:  Geometric constraints on uniformly hyperbolic semigroups


Abstract: We consider semigroups of Mobius transformations that exhibit strong contracting properties on the boundary of the hyperbolic plane. These objects are called uniformly hyperbolic semigroups and appear naturally in the theory of hyperbolic dynamics. The aim of this talk is to present conditions on the generators of a semigroup that guarantee uniform hyperbolicity. These conditions are geometric in nature and are inspired by similar results on Fuchsian groups. 


Speaker: Malavika Mukundan, University of Michigan

Title: Dynamical approximation of entire functions

Abstract: Postsingularly finite holomorphic functions are entire functions for which the forward orbit of the set of critical and asymptotic values is finite. Motivated by previous work on approximating entire functions dynamically by polynomials, we ask the following question:

Given a postsingularly finite entire function f, can f be realised as the locally uniform limit of a sequence of postcritically finite polynomials?

In joint work with Nikolai Prochorov and Bernhard Reinke, we show how we may answer this question in the affirmative 


Speaker: David Martí-Pete, University of Liverpool

Title: Counterexamples to Eremenko's conjecture 

Abstract: The escaping set of an entire function consists of the points of the complex plane whose iterates tend to infinity. For a polynomial, the escaping set is an open neighbourhood of infinity, but for a transcendental entire function, this set is more complicated from a topogical point of view. In 1989, Eremenko proved that the escaping set of a transcendental entire function is never empty and the connected components of its closure are all unbounded. He then conjectured that the components of the escaping set itself are also unbounded. This is known as Eremenko's conjecture and motivated a lot of the research in transcendental dynamics in the recent years. Last year, together with Rempe and Waterman, we proved that the escaping set of a transcendental entire function can have bounded components, which may even be a singleton. Moreover, we proved that every full compact set is a component of the escaping set of some transcendental entire function. In this talk I will discuss the properties of the escaping set and give a sketch of our construction. 


Speaker: Scott Schmieding, Penn State

Title: Stabilized automorphism groups

Abstract: The automorphism group of a topological system consists of all homeomorphisms of the base space which commute with the base dynamics. Recently, Yair Hartman, Bryna Kra, and myself introduced a notion of stabilization for these groups. I'll give some background on this area, discuss some recent work involving something called local P entropy, and some open problems.


Speaker: Bryan Ceasar Felipe, Ateneo de Manila University

Title: Random Substitutions in Higher Dimensions

Abstract: We present a generalization of random substitutions to higher dimensions. In particular, we define a random version of the rectangular-preserving substitutions and digit substitutions. We then discuss the associated subshifts with these random substitutions and present some dynamical properties of these systems by generalizing previous results on one dimensional random substitutions. Moreover, we talk about the existence of periodic points in the subshifts associated with rectangular-preserving random substitutions.


Speaker: Dan Rust, Open University (slides)

Title: Combinatorics of Set-valued Substitutions

Abstract: Symbolic dynamics is a subject that benefits from many areas of mathematics, including topology, ergodic theory, operator theory, number theory and combinatorics. There has been a recent explosion in literature on 'set-valued' or 'random' substitutions, which are like substitutions, but instead of mapping a letter to a single word, it is mapped to an element of a finite set of words according to a non-deterministic generating rule. The language generated by a set-valued substitution can be huge. One way to quantify the size of the language is via the entropy of the system. Another way is to count the number of 'periodic blocks' that the language admits. It's therefore important to develop techniques for counting these quantities and establishing growth rates. In this talk, I'll report on the current state of the art for tackling these problems.


Christmas break


Speaker: Juan Marshall-Maldonado, Aix-Marseille Université

Title: Lyapunov exponents for the spectral cocycle on bijective binary substitutions.

Abstract: The spectral cocycle is an extension of the Rauzy-Veech cocycle, introduced by A. I. Bufetov et B. Solomyak. It is motivated by the study of twisted Birkhoff sums, which have been the focus of many recent works on the spectral measures of dynamical systems.

We study the behavior of the top Lyapunov exponents of the spectral cocycle associated to bijective (in particular, of constant length) substitutions on two letters ; and their topological subshift factors. We prove that for every topological factor coming from a substitution the top Lyapunov exponent does not increase. We also give an explicit sub-exponential deviation from the expected exponential growth of the spectral cocycle. 


Speaker: Professor Marco Abate, Università di Pisa

Title: Random iteration on hyperbolic Riemann surfaces

Abstract: In this talk we shall describe the asymptotic behaviour of sequences of functions obtained by iteratively left-composing or right-composing holomorphic self-maps of a hyperbolic Riemann surface X. We shall consider in detail two cases: when the holomorphic self-maps to be composed have values in a Bloch subdomain of X; and when the holomorphic self-maps to be composed are sufficiently close to a given self-map. This is a joint work with Argyrios Christodoulou.


Speaker: Roger Thompson, Open University (slides)

Title: The Jeandel-Rao Aperiodic Wang Tilings of the Plane

Abstract: In 2015, Jeandel and Rao proved that all tilesets with less than 11 tiles are finite or aperiodic. Using transducers in a brute force computer search, they found 25 candidate aperiodic tilesets with 11 tiles, and proved the aperiodicity of two of them. Labbé determined the underlying structure of one of them. Exploiting this work, and using a different technique for generating large patches, this has enabled the structure of the other, and some of the remaining candidates to be revealed. The technique also greatly simplifies proof of the finiteness of Jeandel and Rao's 10 tile tileset T_h.


Speaker: Phil Rippon, Open University

Title: The Borel–Cantelli lemmas and applications in complex dynamics

Abstract:  In this largely expository talk we discuss the Borel–Cantelli lemmas and various extensions and generalisations of these, and we describe some ways in which these lemmas have been applied in complex dynamics, for example, in joint work with Anna Benini, Vasso Evdoridou, Nuria Fagella and Gwyneth Stallard.


Speaker: Sabrina Kombrink, University of Birmingham

Title: Parallel volumes and spectral counting functions for fractals

Abstract: We will discuss two applications of dynamical renewal theory in geometry. Firstly, we will look at asymptotic expansions of parallel volumes of fractal sets – aiming at introducing notions of “fractal volume”, “fractal surface area”, “fractal curvatures”. Secondly, by finding higher order asymptotic terms of eigenvalue counting functions of the Dirichlet Laplacian on domains with a fractal boundary, we will address the question which geometric features of a drum with a fractal boundary one can hear when listening to its sound. The proofs rely on spectral properties of Ruelle-Perron-Frobenius operators.


Speaker: Bogusława Karpinski, Warsaw University of Technology

Title: On the dimension of  escaping sets for non-autonomous iteration of exponentials

Abstract:   In this talk we present general estimates for the Hausdorff and packing dimension of sets of points which escape to infinity at a given rate under non-autonomous iteration of exponentials. In particular we  discuss some conditions which guarantee that the Hausdorff and packing dimensions achieve given values in the interval [1,2].  The talk is based on a joint work with Krzysztof Barański.


Speaker: Margaret Stanier, The Open University (slides)

Title: The convergence of integer continued fractions

Abstract: Fundamental to the theory of continued fractions is the fact that every infinite continued fraction with positive coefficients converges. This is not the case, however, if the coefficients are not necessarily positive. We use a geometric representation of an integer continued fraction to find a simple test which determines whether such a fraction converges or diverges.  This is joint work with Ian Short.



Summer break


Aperiodic Tilings

    A meeting and mathematical art exhibition in honour of Uwe Grimm

    https://www.open.ac.uk/aperiodic-tilings-2022


Speaker: Thomas Richards, Mathematics Institute, University of Warwick

Title: Monodromy and complex Hénon maps

Abstract: Blanchard, Devaney, and Keen proved that loops in the shift locus of degree $d$ polynomials induce automorphisms of the one-sided shift of $d$ symbols. Hubbard conjectured that an analogous result holds in H\'enon parameter space. In this talk, I will discuss this conjecture and some experimental observations we have made.


Speaker: Joanna Kulaga-Przymus, Nicolas Copernicus University, Torun-Poland

Title: Around Sarnak's conjecture

Abstract: I will give an introduction to the subject of Sarnak's conjecture (2010) and talk about the resulting interplay of number theory and dynamical systems, including some recent related results.


Speaker: Vasiliki Evdoridou, Faculty of STEM, The Open University and MSRI, Berkeley.

Title: The Denjoy-Wolff set and wandering domains

Abstract: In the iteration of holomorphic self-maps of the unit disc the dynamical behaviour of points in the disc is determined by the well-known Denjoy-Wolff theorem. Specifically for inner functions, there is a remarkable dichotomy in the behaviour of boundary points. Motivated by questions on the boundary behaviour of wandering domains, i.e. Fatou components that are not eventually periodic, we extend the notion of the Denjoy-Wolff point to sequences of holomorphic functions between simply connected domains, and define the Denjoy-Wolff set; those points on the boundary whose images have the same limiting behaviour as the images of all interior points. Moreover, we study the aforementioned dichotomy in this more general setting. We will focus on the special case of simply connected wandering domains of transcendental entire functions, and see how these results help us relate the internal behaviour with the behaviour of boundary points. This is joint work with A.M. Benini, N. Fagella, P. Rippon and G. Stallard.


Speaker: Simon Baker, School of Mathematics, University of Birmingham

Title: Normal numbers and self-similar measures.

Abstract: A real number x is said to be normal in base b if the sequence (b^n x) is uniformly distributed modulo one. In this talk I will discuss a recent result which states that for a self-similar measure and an integer b, if the self-similar measure and b satisfy a suitable arithmetic assumption, then almost every x is normal in base b with respect to the self-similar measure. This is a joint work with Amir Algom and Pablo Shmerkin.


Speaker: Alvaro Bustos-Gajardo, Faculty of STEM, The Open University

Title:  B-free number-theoretical shift spaces and their symmetries

Abstract: We discuss shift spaces constructed from indicator functions of number-theoretically defined sets, focusing mostly on their factorisation properties (examples include the d-dimensional shift of visible lattice points and the family of k-free shift spaces coming from a finite field extension of Q). These subshifts exhibit interesting (and, from certain perspectives, unusual) properties, including the combination of high complexity (positive entropy) with symmetry rigidity (the automorphism group is “essentially trivial”, containing only shift maps). We will talk mostly about two-dimensional examples, focusing our discussion on the extended symmetry group, which is a mild generalisation of the automorphism group which exhibits a wide variety of interesting behaviours in this context.


Speaker: Ana Rodrigues, College of Engineering, Mathematics and Physical Sciences, University of Exeter

Title: Dynamics of piecewise isometries

Abstract: In this talk I will discuss some features of the dynamics of Piecewise isometries (PWIs) which are higher dimensional generalizations of one dimensional IETs, defined on higher dimensional spaces and Riemannian manifolds. In particular, I will introduce the concept of embedding of an IET into a PWI, some particular renormalization scheme and if time allows, the proof of existence of invariant curves for PWIs.


Speaker: Nicolai Edeko, Institut für Mathematik, Universität Zürich 

Title: Isometric factors of dynamical systems on locally path-connected spaces

Abstract: Given a dynamical system, heuristically, a factor is simpler than the original system. But what does this mean, concretely? For example, if a dynamical system has a certain topological/algebraic structure, is it true that its factors must have a simpler topological/algebraic structure? This talk contributes to results in this vein for the special case of isometric factors: We will discuss why isometric factors for transitive homeomorphisms on compact manifolds are compact abelian Lie groups and infer some spectral-theoretic consequences.


Speaker: Fabrizio Bianchi, Laboratoire Paul Painlevé, Lille

Title: Holomorphic motions of Julia sets: dynamical stability in one and several complex variables

Abstract: In this talk we discuss stability of holomorphic dynamical systems under perturbation. In dimension 1, the theory is now classical and is based on works by Lyubich, Mané-Sad-Sullivan and DeMarco. I will review this theory and present a recent generalisation valid in any dimension. Since classical 1-dimensional techniques no longer apply in higher dimensions, our approach is based on ergodic and pluripotential methods.


Speaker: Olga Lukina, University of Vienna

Title: Stabilizers in group Cantor actions and measures

Abstract: Given a countable group G acting on a Cantor set X by transformations preserving a probability measure, the action is essentially free if the set of points with trivial stabilizers has full measure.  In this talk, we consider actions where no point has trivial stabilizer, and investigate the properties of the points with non-trivial holonomy. We introduce the notion of a locally non-degenerate action, and show that if an action is locally non-degenerate, then the set of points with trivial holonomy has full measure in X. We discuss applications of this work to the study of invariant random subgroups, induced by actions of countable groups. This is joint work with Maik Gröger. 


Speaker: Sebastien Biebler, Université de Paris VII, MSRI 

Title: Wild holomorphic dynamical systems (joint work with Pierre Berger, CNRS, Sorbonne University) 

Abstract:  In the 60s, in a mathematical optimistic movement aiming to describe a typical dynamical system, Smale conjectured the density of uniform hyperbolicity in the space of C^r-diffeomorphisms f of a compact manifold M. In 1974, a student of Smale, Newhouse, discovered an extremely complicated new phenomenon, resulting in an obstruction to Smale's conjecture. Specifically, Newhouse showed the existence of (nonempty) open sets U  of C^2-diffeomorphisms of a surface M such that a generic map f in U has infinitely many attracting periodic points. In particular, the statistical behavior of such systems can not be described in a satisfying way with a finite number of measures.

In this talk, I will first define precisely the Newhouse phenomenon. Then, I will discuss a joint work with Pierre Berger whose proof is based on the Newhouse phenomenon. We show that there exist polynomial automorphisms of C^2 with a wandering Fatou component. This result contrasts with a celebrated theorem of Sullivan who proved in the 80s that any rational map of the Riemann sphere does not have such wandering Fatou components. We also study the statistical behaviour of orbits of points inside the wandering component, and we show that it is very difficult to describe, namely historic with high emergence. 


Speaker: James Walton, School of Mathematical Sciences, University of Nottingham

Title: Compact substitutions

Abstract: In extending the study of symbolic substitutions from finite to infinite alphabets, one encounters several obstacles to generalising most of the standard theory. So instead of considering substitutions on arbitrary alphabets, we choose to retain some extra structure from the finite case by demanding that the alphabet carries a compact Hausdorff topology for which the substitution is continuous. Several notions from the classical case can be naturally extended to this setting, such as of a substitution being primitive. Surprisingly, primitivity is no longer sufficient to ensure unique ergodicity of the associated shift space. But we may find conditions which imply unique ergodicity, and which may be verified on a wide range of examples. In place of Perron-Frobenius theory from the finite case, we make use of the theory of positive operators on Banach spaces. This is joint with work Neil Mañibo and Dan Rust.

  

Speaker: Weiwei Cui, Centre for Mathematical Sciences, Lund University

Title: Hausdorff dimension of Julia and escaping sets of meromorphic functions

Abstract:  We discuss recent results on the Hausdorff dimension of Julia and escaping sets for meromorphic functions with finitely many singular values (i.e., Speiser functions). This is based on recent joint works with Magnus Aspenberg (Lund) and respectively Walter Bergweiler (Kiel).


Christmas break


Speaker: Toby Taylor Crush, Loughborough University

Title: Digit distribution in random continued fractions 

Abstract: The distribution of digits in the continued fraction representation of a number was first discovered by Gauss, we approximate the distribution for random continued fractions with arbitrary accuracy by first approximating the long term statistics of the shift map on the digits of our numerical representation using information from the deterministic case.


Speaker: Anna Jové Campabadal, University of Barcelona

Title: Boundary orbits in Fatou components and the rigid example z+exp(-z) 

Abstract: In this talk, we discuss the dynamics of the particular example f(z)=z+exp(-z), which presents infinitely many invariant doubly-parabolic Baker domains. In particular, we are interested in describing the topology of the boundary and the dynamics on it. This includes characterizing the accesses to infinity, periodic points and the escaping set. To do so, first some results on the ergodic properties of boundary maps of inner functions will be presented.


Speaker: Ibai Aedo, The Open University

Title: Forward limit sets of substitution semigroups

Abstract: We introduce substitutions of alphabets and explore their behaviour under iteration. This theory is enriched by considering collections of substitutions that generate semigroups of substitutions. Associated to each semigroup is a geometric object called a forward limit set, comprising a collection of infinite words in the alphabet. We will discuss properties of forward limit sets such as their size and their relationship to limit points of certain sequences in substitution dynamics known as S-adic sequences 


Speaker: Neil Dobbs, University College Dublin

Title: Lower bounds on the dimension of some quadratic Julia sets.

Abstract: Within hyperbolic components, the Hausdorff dimension of quadratic Julia sets varies analytically. On the boundary of the Mandelbrot set, on the other hand, it varies discontinuously. Indeed, there is a residual set of parameters (in the boundary) where the dimension is 2, and a full harmonic measure set where it is less than 2. For the quadratic  z^2 -2, the Julia set is an interval and hence has dimension 1. We shall show how to obtain strong lower bounds in a neighbourhood of this map.



Speaker: Philipp Gohlke, Bielefeld University

Title: The Thue-Morse measure: multifractal analysis and superpolynomial scaling

Abstract: The Thue-Morse measure is a paradigmatic example of a singular continuous measure that arises from a system of aperiodic order. Its properties have been studied extensively in the past, including a partly heuristic multifractal analysis in the mathematical physics literature.

The aim of a multifractal analysis is to obtain a detailed understanding of the scaling behaviour of the measure around individual points. We revisit this analysis in the framework of the thermodynamic formalism, interpreting the Thue-Morse measure as an equilibrium measure for a potential with a singularity. This singularity gives rise to a superpolynomial scaling behaviour around dyadic points and produces a pathological feature in the multifractal spectrum.


Speaker: George Kenison,  TU Wien

Title: On minimality and positivity for second-order holonomic sequences.

Abstract: An infinite sequence <u_n> of real numbers is holonomic if it satisfies a linear recurrence relation with polynomial coefficients. Such a sequence is positive if each u_n >= 0, and minimal if, given any other linearly independent sequence <v_n> satisfying the same recurrence relation, the ratio u_n/v_n converges to 0 as n tends to infinity.

In recent work, the speaker and collaborators establish a Turing reduction of the problem of deciding positivity of second-order holonomic sequences to that of deciding minimality of such sequences.  More specifically, we give a procedure for determining positivity of second-order holonomic sequences that terminates in all but an exceptional number of cases, and we show that in these exceptional cases positivity can be determined using an oracle for deciding minimality.

This is joint work with O. Klurman, E. Lefaucheux, F. Luca, P. Moree, J. Ouaknine, M.A. Whiteland, and J. Worrell.


Speaker: Anna Zdunik, University of Warsaw

Title: Conformal measures and Ruelle’s property for entire and meromorphic maps.

Abstract: In this talk, I will discuss the questions of thermodynamic formalism in the context of transcendental dynamics: topological pressure, existence of conformal measures on the radial Julia set, Bowen’s formula and Ruelle’s property. I will present some results obtained in my joint papers with Mariusz Urbański, Bogusława Karpińska, Krzysztof Barański and Volker Mayer.


Speaker:  Dan Paraschiv, University of Barcelona

Title: Achievable connectivities of Fatou components for a family of singular perturbations 

Abstract: By adding a perturbation at a singular point of a holomorphic map, even richer dynamics may be obtained (see the Trichotomy Theorem by Devaney, Look, and Uminsky). For a family of perturbation maps, Canela has proven the existence of a rational map such that, in one dynamical plane corresponding to a parameter, the Fatou components have arbitrarily large connectivity . We generalise this result to a larger family of maps and also precisely compute all the achievable connectivities.


Speaker:  Charlene Kalle,  Leiden University

Title: Critical intermittency in random dynamical systems

Abstract: We consider a type of intermittent behaviour for random dynamical systems, where on the one hand the dynamics is attracted to some superattracting fixed point, but on the other hand orbits close to this fixed point can be mapped to the neighbourhood of some repelling fixed point. We discuss the existence and the properties of absolutely continuous invariant measures for random systems on the interval admitting this type of critical intermittency. 


Speaker: Dave Sixsmith, The Open University

Title: The maximum modulus set of an entire function

Abstract: We discuss (very partial) progress towards answering the problem of giving necessary and sufficient conditions for a set $S \subset \mathbb{C}$ to be the maximum modulus set of an entire function f.


Speaker: David Martí -Pete, University of Liverpool

Title: The dimension of the boundaries of basins of attraction of hyperbolic entire maps 

Abstract: Let f be a transcendental entire function, and let U be a Fatou component of f. It follows from a result of Baker that the Hausdorff dimension of the boundary of U is larger than or equal to 1, and Bishop recently constructed an entire function with a multiply connected wandering domain whose boundary has Hausdorff dimension equal to 1. Thus, it is natural to ask for which Fatou components the boundary can have Hausdorff dimension 1.

We study the case that f is in the Eremenko-Lyubich class and U is an invariant basin of attraction of f. For such maps, Stallard proved that the Hausdorff dimension of the Julia set is strictly larger than one. Suppose that the set of singular values in U is a compact subset of U and, in the case that U is unbounded, assume that the degree of f in U is infinite. Then the Hausdorff dimension of the boundary of U is strictly larger than 1. This generalizes a previous result of Barański, Karpińska and Zdunik concerning the exponential family. 

Recall that for polynomials, Zdunik proved that if U is an invariant basin of attraction, then the Hausdorff dimension of the boundary of U is strictly greater than 1 unless it is an analytic Jordan curve or an arc. Then Przytycki extended this result to all rational functions. Brolin proved that, for rational functions, the only cases where the boundary of a basin of attraction can be analytic are when it is a circle or an arc of a circle. We prove that if U is a bounded invariant basin of attraction of a transcendental entire function f, then the boundary of U is not an analytic Jordan curve or an arc. 

This is a joint work with K. Barański, B. Karpińska, L. Pardo-Simón and A. Zdunik.



Speaker: Dimitrios Ntalampekos, Stony Brook University

Title: David homeomorphisms and applications in mating and removability

Abstract:  The main object in this talk will be the mating of piecewise (anti-)analytic dynamical systems of the unit disk. While quasiconformal maps can be used for the mating of two hyperbolic dynamical systems, they are insufficient for mating a hyperbolic dynamical system with a parabolic one. Instead, we achieve the mating using the notion of a David homeomorphism, which is a generalization of a quasiconformal homeomorphism that allows unbounded quasiconformal dilatation. The main theorem that we will discuss provides extensions of a general class of dynamically defined circle homeomorphisms to David homeomorphisms of the unit disk. An implication of this theory is that limit sets of a certain class of Kleinian reflection groups (called necklace reflection groups) are conformally removable. The talk is based on joint work with Misha Lyubich, Sergei Merenkov, Sabyasachi Mukherjee, and Christina Karafyllia. 


Speaker:  Rhiannon Dougall, University of Bristol

Title: Comparison of entropy for infinite covering manifolds, and group extensions of subshifts of finite type

Abstract:. A classical example of an Anosov flow would be the geodesic flow associated to a compact hyperbolic manifold M. The periodic orbits are then closed geodesics in M, and this topic has a rich history. In general Anosov flows are not so well behaved, there may be infinitely many periodic orbits in a free homotopy class, in contract to geodesic flows. Nevertheless one has results on the asymptotic number of periodic orbits up to period T. In this talk we will discuss the problem of counting periodic orbits in infinite covering manifolds, where we relate the exponential growth rate of periodic orbits in the cover to properties of the covering group. Such results are rooted in analogous statements for group extensions of subshifts of finite type -- I will spend some time motivating this setting too. Featuring joint work with Richard Sharp.

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Speaker:  Athanasios Tsantaris, University of Nottingham

Title: Dynamics of Zorich maps

Abstract: In the theory of one dimensional holomorphic dynamics, one of the most well studied families of maps is the exponential family $E_\lambda(z):=\lambda e^z$. Zorich maps are the quasiregular higher dimensional analogues of the exponential map on the plane.  For the exponential family $E_\lambda(z):=\lambda e^z$, $\lambda>0$ it is generally well known that for $0<\lambda\leq 1/e$ the Julia set of $E_\lambda$ is a "Cantor Bouquet" while for $\lambda>1/e$ the Julia set is the entire complex plane. In this talk we will discuss how this dichotomy and many other facts about the exponential family generalize to the higher dimensional setting of Zorich maps 

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Speakers: Farhana Pramy and Ben Mestel , The Open University

Title: A computer-assisted proof of magnetic-field growth for the Stretch-Fold-Shear model of a kinematic dynamo 

Abstract: The Stretch-Fold-Shear family $S_\alpha$ is a one-parameter family of linear operators acting on complex-valued functions $c(x)$, for $x \in [-1,1]$, and parametrised by a real parameter $\alpha \ge 0$. The family arises from a stylized model of magnetic field growth in kinematic dynamo theory that was developed by Andrew Gilbert and for which an eigenvalue of modulus greater than 1 corresponds to magnetic field growth.

 In this talk we describe a computer-assisted proof of the existence of an eigenvalue of $S_\alpha$ of modulus greater than 1 for $\alpha$ in the range $\pi/2 < \alpha \le 5$, thereby partially proving a conjecture of Gilbert.



Speaker: James Waterman, University of Liverpool

Title: Docile entire functions

Abstract:  Several important problems in complex dynamics are centered around the local connectivity of Julia sets of polynomials and of the Mandelbrot set. Importantly, when the Julia set of a polynomial is locally connected, the topological dynamics of the map can be completely described as a quotient of a power map on the circle.

 Local connectivity of the Julia set is less significant for transcendental entire functions. Nevertheless, by restricting to a class of transcendental entire functions, known as docile functions, we obtain a similar concept by describing the topological dynamics as a quotient of a simpler disjoint-type map. We will discuss the notion of docile functions, as well as some of their properties. This is joint work with Lasse Rempe.

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Speaker: Anna Miriam Benini, University of Parma

Title: Baker domains in one and two variables

Abstract:  For  polynomial maps  on the complex plane $\mathbb{C}$ there is only one way in which orbits can converge to infinity: they have to belong to the attracting basin of infinity, which can be seen as a superattracting fixed point.  For transcendental maps infinity is an essential singularity, and there are many ways in which orbits can converge to infinity while belonging to a periodic Fatou component.  Such components are called  Baker domains, and can sport several different dynamical features. 

In $\mathbb{C}^2$ we call a Fatou component \emph{escaping} if the orbits within  converge to the line at infinity (we consider the projective space as compactification of $\mathbb{C}^2$ ). The picture  is somewhat similar as the one-dimensional case: for polynomial automorphisms of  $\mathbb{C}^2$,  escaping orbits belong to a Fatou component which can be seen as an attracting basin of a point on the line at infinity, while for transcendental automorphisms, many more possibilities for escaping Fatou components open up both from the geometrical and from the dynamical point of view. We will describe a few examples and  present some of the many possible open questions that arise. This talk includes results obtained with Arosio, Fornaess, Peters and with Saracco, Zedda.



Speaker: Tom Kempton, University of Manchester

Title: Bernoulli Convolutions and Measures on the Spectra of Algebraic Integers 

Abstract: Given an algebraic integer beta and alphabet A = {-1,0,1}, the spectrum of beta is the set

 

   \Sigma(\beta) := \{ \sum_{i=1}^n a_i \beta^i : n \in \mathbb{N}, a_i \in A \}.


In the case that beta is a Pisot number (a type of algebraic number) one can study the spectrum of beta dynamically using substitutions or cut and project schemes, and this allows one to see lots of local structure in the spectrum. 

In this talk we will define a random walk on the spectrum of beta and show how, with appropriate renormalisation, this leads to an infinite stationary measure on the spectrum. This measure has local structure analagous to that of the spectrum itself. We will also explain some motivation/applications of this measure to fractal geometry.

In this talk we will touch on some ideas from number theory and some ideas from fractal geometry but no previous knowledge will be assumed.

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Speaker: Mareike Wolff, Christian-Albrechts-Universität zu Kiel

Title: A class of Newton maps with Julia sets of Lebesgue measure zero

Abstract:  Let g(z) = \int_0^{z} p(t) exp(q(t))dt +c with polynomials p and q, and let f be the function from Newton's method for g. Under suitable assumptions, we can show that the Julia set of f has Lebesgue measure zero. Together with a theorem by Bergweiler, this implies that f^n(z) converges to zeros of g almost everywhere in the complex plane if this is the case for each zero of g'' that is not a zero of g or g'. We will discuss these results and sketch the proof of our result which is based on a more general theorem on Julia sets of Lebesgue measure zero.

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Speaker: Clemens Müllner, TU Wien

Title: Multiplicative automatic sequences

Abstract: It was shown by Mariusz Lemańczyk and the author that automatic sequences are orthogonal to bounded and aperiodic multiplicative functions.

This is a manifestation of the disjointedness of additive and multiplicative structures. We continue this path by presenting in this talk a complete classification of complex-valued sequences which are both multiplicative and automatic. This shows that the intersection of these two worlds has a very special (and simple) form. This is joint work with Mariusz Lemańczyk and Jakub Konieczny.

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Speaker: Clifford Gilmore,  University College Cork

Title: Dynamics of Weighted Composition Operators on Fock Spaces

Abstract:  Linear dynamics has been a rapidly evolving area of research since the early 1990s. I will begin by introducing the notion of hypercyclicity and by recalling some basic examples to illustrate the dynamics of continuous linear operators in the setting of infinite-dimensional Hilbert/Banach spaces. 

On the other hand, the study of weighted composition operators acting on spaces of analytic functions has recently been the subject of much research activity. In particular, characterisations of the bounded and compact weighted composition operators acting on Fock spaces were identified by, amongst others, Ueki (2007), Le (2014), and Tien and Khoi (2019).  

In this talk I will examine some recent results that give explicit descriptions of bounded and compact weighted composition operators acting on Fock spaces. This allows us to prove that Fock spaces do not support supercyclic weighted composition operators. This is joint work with Tom Carroll (UCC).


Speaker: Liviana Palmisano,  Durham University

Title: Attractors and their stability

Abstract: One of the fundamental problems in dynamics is to understand the attractor of a system, i.e. the set where most orbits spent most of the time. As soon as the existence of an attractor is determined, one would like to know if it persists in a family of systems and in which way i.e. its stability. Attractors of one dimensional systems are well understood, and their stability as well. I will discuss attractors of two dimensional systems, starting with the special case of Henon maps. In this setting very little is understood. Already to determine the existence of an attractor is a very difficult problem. I will survey the known results and discuss the new developments in the understanding of attractors, coexistence of attractors and their stability for two dimensional dynamical systems.


Speaker: Petra Staynova, University of Derby

Title: From Ramsey Theory to Combinatorics on Words: How a forgotten proof helped find long arithmetic progressions

Abstract: The original proofs of classic Ramsey-theoretic results, such as van der Waerden's theorem, have been forgotten and replaced by more elegant and mysterious proofs via ultrafilters and the Stone-Czech compactification of the integers. However, there is still good value to be found in the original proofs. In particular, van der Waerden's proof of his theorem about the existence of arbitrarily long monochromatic arithmetic progressions is both beautifully combinatorial and intuitive. It gives a concrete construction of the arithmetic progressions, and an upper bound on 'how far you have to look before you find a monochromatic arithmetic progression'. We can use van der Waerden's ideas to give an elegant visual proof of a combinatorially difficult result about arithmetic progressions within the Thue-Morse word. Furthermore, we can show that the progression we find via this method is, in some sense, the longest possible. This is joint work with Ibai Aedo, Uwe Grimm, and Yasushi Nagai.

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Speaker: Trevor Clark,  The Open University

Title: Real dynamics from the complex point of view

Abstract: Tools from complex analysis have played a crucial role in many results in one-dimensional real dynamics, for example, in monotonicity of entropy, density of hyperbolicity and exponential convergence of renormalization. In this talk, I will discuss some of these tools, and how they can be used.

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Speaker: Leticia Pardo Simón, IMPAN 

Title: Criniferous entire functions and Cantor bouquet Julia sets 

Abstract:  It is known for a large number of transcendental entire functions with bounded singular set that every escaping point can be connected to infinity by a curve of escaping points, now often called (Devaney) hairs.  When this is the case, we say that the function is criniferous. Although not all functions with bounded singular set are criniferous, those with finite order of growth are, and, in some special cases, their Julia set is a collection of hairs forming a topological object known as Cantor bouquet. In this talk, we describe a new class of criniferous functions and explore their relation to Cantor bouquets. This is joint work with L. Rempe. 

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Speaker: Neil Mañibo ,  Bielefeld University

Title: On measures arising from regular sequences

Abstract: Regular sequences are generalisations of automatic sequences. Unlike the latter, they are not necessarily bounded and hence the usual spectral approach via correlation measures does not apply to them in general. Nevertheless, they still satisfy certain recursive properties. In this talk, we will discuss how one can associate a sequence of measures to these objects and we will provide conditions which ensure the vague convergence of these approximants. We will also mention some properties of the limit measure (spectral type, local dimension), and some open questions. (Joint work with Michael Coons and James Evans)

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Speaker: Dan Rust,  The Open University

Title: Random substitutions: their dynamics and Rauzy fractals

Abstract: Quasicrystals and aperiodic tilings are non-periodic patterns which still exhibit a remarkable regularity, characterised by non-trivial pure point spectrum. In a sense, they are globally very well-ordered. The question then is how locally disordered can they be? I will first review 1 dimensional substitution tilings, their dynamical systems and their Rauzy fractals. I will then introduce 'random substitutions' which are locally scrambled versions of substitutions. Random substitutions allow us to construct hierarchical tilings which are still globally well-ordered, but are locally disordered, characterised by positive entropy. We will see that these tilings still have Rauzy fractals that can tell us something about their dynamical systems but with a much richer metric structure. If time permits, I'll show how we can use these methods to test their mixing properties.

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Speaker: Kurt Falk,  Christian-Albrechts-Universität zu Kiel 

Title: Dimension gaps for limit sets of Kleinian groups 

Abstract:  In this talk I will give a brief survey of results on dimension gaps for limit sets of geometrically infinite Kleinian groups. I will concentrate on an important notion from geometric group theory, amenability, as a criterion for the existence of such gaps. 

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Speaker: Luna Lomonaco, IMPA

Title: Mating quadratic maps with the modular group 

Abstract:  Holomorphic correspondences are polynomial relations P(z,w)=0, which can be regarded as multi-valued self-maps of the Riemann sphere  (implicit maps sending z to w). The iteration of such multi-valued map generates a dynamical system on the Riemann sphere (dynamical system which generalise rational maps and finitely generated Kleinian groups). We consider a specific 1-(complex)parameter family of (2:2) correspondences F_a (introduced by S. Bullett and C. Penrose in 1994),  which we describe dynamically. In particular, we show that for every a in the connectedness locus M_{\Gamma}, this family is a mating between the modular group and rational maps in the family Per_1(1), and we develop for this family a complete dynamical theory which parallels the Douady-Hubbard theory of quadratic polynomials. This is joint work with S. Bullett.

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Speaker: Wolfgang Steiner, IRIF Paris 

Title: Multidimensional continued fraction algorithms and symbolic codings of toral translations 

Abstract: The aim of this lecture is to construct good symbolic codings for translations on the d-dimensional torus that enjoy many of the beautiful properties of Sturmian sequences (as for instance low complexity and bounded remainder sets of any scale). Inspired by Rauzy's approach, we obtain such codings for almost all directions (in dimensions 2 and 3) by using sequences of substitutions that come from multidimensional continued fraction algorithms. This is joint work with Valérie Berthé and Jörg Thuswaldner. 

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Speaker: Maria Kourou,  Julius-Maximilians-Universität Würzburg 

Title: Harmonic measures and semigroups of holomorphic functions

Abstract: Let  (φ_t)_{t>=0} be a semigroup of holomorphic self-maps of the unit disk D with Denjoy-Wolff point  τ  on the unit circle.  Suppose K is a compact subset of the unit disk D.  Its image φ_t(K) is also a compact set in D that shrinks to  τ, as t tends to infinity. The aim is to examine the way it approaches the boundary and how its size changes, as t increases. In the present talk, we focus on results that are related with the harmonic measure with respect to φ_t(K). Furthermore, we observe that the asymptotic behaviour of certain harmonic measures provides us with information on the type of the semigroup.

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Speaker: Yasushi Nagai, The Open University

Title: Arithmetic progressions, pure point diffraction and limit-periodicity for self-affine tilings 

Abstract: The repetitions of patterns in a tiling is an important topic in the study of tilings as models of quasicrystals. Arithmetic progressions in tilings are one type of repetition. On the other hand, there are several interpretations of a vague claim that a tiling is "ordered". Pure point diffraction and limit periodicity are such examples. In this talk we discuss the relations among arithmetic progressions, pure point diffraction and limit-periodicity for a class of tilings called self-affine tilings.

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Speaker: Xavier Jarque, Universitat de Barcelona

Title: The secant map as a dynamical system in R^2 

Abstract: We deal in this work with the plane dynamical system given by the secant map applied to a polynomial p of degree d with real coefficients.  We present some properties of the basins of attraction of the fixed points of the plane secant map associated to the real (simple) zeros of p. The main tool is to apply a general theory for certain rational maps of the plane. There is a natural, although I guess difficult, extension of this work to holomorphic two-dimensional dynamical systems in C^2 (or{CP}^2).  Joint work with T. Garijo and L. Gardini.

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Speaker: Reem Yassawi, The Open University

Title: Recognizability for sequences of morphisms 

Abstract:  In this talk a symbolic dynamical system (X, T) is one where X is a Cantor space, typically a closed subset of bi-infinite sequences on a finite alphabet A, and T is the left shift map. The set of these symbolic dynamical systems is large. Within that set, there are many that are actually symbolic representations of dynamical systems where the phase space is the unit interval, and dynamicists like to identify them.

I’ll talk about the family of cutting-and-stacking transformations, which are prototypical transformations on the unit interval that preserve Lebesgue measure μ. Each such transformation ([0, 1], S, μ) yields a symbolic dynamical system called an S-adic shift (X, T, μ ̃). The space X is defined by a sequence of morphisms (σ(n)), which map letters in an alphabet A(n + 1) to words on an alphabet A(n). In general, ([0, 1], S, μ) and (X, T, μ ̃) are not the same. There is a map F : (X, T, μ ̃) → ([0, 1], S, μ), which commutes T with S, but the problem is that it is not injective.

Recognizability is a purely combinatorial concept which translates to injectivity of F. In this talk I will survey these notions, and talk about conditions which guarantee recognizability, and give examples where there is no recognizability. This is joint work with Valerie Berthe, Wolfgang Steiner and Jorg Thuswaldner.

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Speaker: Phil Rippon, The Open University

Title: Eremenko's conjecture and the iterated minimum modulus 

Abstract: In 1989, Eremenko conjectured that for any transcendental entire function f, the escaping set I(f)={z:f^n(z)\to\infty as n->\infty} has no bounded components - despite much work this conjecture is still open. This talk will focus on real entire functions of finite order with only real zeros. We show that Eremenko's conjecture holds for such functions f, and moreover I(f) is connected and has a `spider's web' structure, if there exists r>0 such that the iterated minimum modulus m^n(r)->\infty as n ->\infty. Here m(r)=\min_{|z|=r}|f(z)|. We discuss examples of families of entire functions for which this iterated minimum modulus condition does, and does not, hold. This is joint work with Dan Nicks and Gwyneth Stallard. 

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Speaker: Argyrios Christodoulou, University of Surrey

Title: The Hausdorff dimension of self-projective fractals

Abstract: In this talk we discuss iterated function systems (IFS) on real projective space. In particular, for IFS that satisfy an exponential separation condition and are sufficiently contractive, we give an explicit formula for the Hausdorff dimension of their attractor, in terms of the minimal root of their pressure function. This generalises a recent result of  Solomyak and Takahashi. Joint work with Natalia Jurga.

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