Mondays at 2pm
Frank Adams 1, Alan Turing Building
Contacts Donald Robertson Yotam Smilansky
Extensions and applications to Sullivan's dictionary
Julia Münch
University of Liverpool
For generalising complex dynamics to higher real dimensions it is natural to consider quasi-regular mappings. The subfamily of uniformly quasi-regular mappings is particularly well understood but it is difficult to find interesting examples. I will be talking about a result obtained joint with Daniel Meyer; we showed that one can extend a certain class of holomorphic functions (expanding Thurston maps) on the Riemann sphere S^2 to a uniformly quasi-regular mapping F defined on a subset Ω of R^3 containing S^2 in its interior.
In the second part of the talk we will present two applications of this construction that is motivated by Sullivan’s dictionary. Sullivan’s dictionary stipulates an analogy between objects, conjectures and theorems in complex dynamics and the study of Kleinian groups. We will examine the properties of the extension F with respect to the hyperbolic metric, and show an application that has a counterpart in the theory of Kleinian groups. The aim of the talk is to explain a construction of a space-filling curve that is the boundary of an immersed and severely folded plane.
Discrete spectrum in S-adic substitution dynamical systems via the balanced pair algorithm
Eng-Jon Ong
Queen Mary University of London
Substitutions and S-adic dynamical systems are a natural framework for generating and studying infinite symbolic sequences via the iterative application of substitution rules. For two-letter symbolic dynamical systems arising from a single Pisot substitution, it is known that the associated dynamical system has purely discrete spectrum. In this talk, we extend this result to a class of S-adic systems generated by a finite set of substitutions. Our approach is based on the Balanced Pair Algorithm, and we prove that this algorithm terminates for two-symbol S-adic systems, a key step in establishing pure discrete spectrum in this setting. We also establish the existence of an asymptotic expansion rate for growing words in S-adic systems.Joint dynamics event, details and updates can be found in https://sites.google.com/view/manchesterdynamics/home
One-day ergodic theory meeting (March 30th)
Claire Burrin (Zurich), Joel Moreira (Warwick), Tuomas Sahlsten (Helsinki).
Grimm network in aperiodic order meeting (March 31st)
Andrew Mitchell (Loughborough), Sigrid Grepstad (Norwegian University of Science and Technology), Ram Band (Technion), Rodrigo Treviño (Maryland).
Rachid El Harti
Hassan University
Prime periods on the interval
Gabriel Fuhrmann
Durham University
Given two continuous self-maps f and g on the interval which have all periodic orbits in common (that is, O(x)={x,f(x),...,f^(p-1)(x)} is a p-periodic orbit of f if and only if it is a p-periodic orbit of g but a priori, f may permute the elements of O(x) in a different fashion than g does), it is natural to ask whether f=g on the closure of the periodic points (which is known to coincide with the closure of the recurrent points!). We show this is the case wherever orbits with prime periods are dense. Specifically, we show that mixing interval maps are uniquely determined by (the location of) their periodic orbits. Joint work with Maik Gröger (Jagiellonian University) and Alejandro Passeggi (University of the Republic Uruguay).Hollow wandering domains in quasiregular dynamics
Dan Nicks
University of Nottingham
Quasiregular mappings of n-dimensional Euclidean space generalise holomorphic functions on the complex plane. One can thus study the iteration of quasiregular maps, aiming to uncover parallels and differences to successful theory of complex dynamics. One major difficulty is that, unlike holomorphic functions, quasiregular maps locally distort space by a bounded amount, and the amount of distortion increases with each iteration. After some introduction to quasiregular dynamics, I will discuss some wandering domains that are a quasiregular analogy to multiply-connected Fatou components of transcendental entire functions.The nuclear dimension of groupoid C*-algebras
Christian Bönicke
Newcastle University
Nuclear dimension is a noncommutative generalisation of Lebesgue covering dimension for compact Hausdorff spaces. Finiteness of this dimension plays an important role in the classification programme for simple nuclear C*-algebras. In this talk I will give a brief introduction to this dimension theory and explain an approach for estimating the nuclear dimension for C*-algebras arising from topological dynamical systems.Ergodic theorems for unimodular amenable groups via random walk approach
Runlian Xia
University of Glasgow
The pointwise ergodic theorem for amenable groups was first proved by Lindenstrauss in 2001. Very recently, this result was extended to the noncommutative setting - where the measure space is replaced by a von Neumann algebra equipped with a semifinite normal faithful trace - by Caldihac and Wang. In this talk, we show that the ergodic averages of the action of any unimodular amenable group along certain Følner sequences can be dominated by the Cesàro means of a suitably constructed Markov operator; that is, by the ergodic averages of an integer action. As a consequence, we provide a unified and simpler proof of the two results in the commutative and noncommutative setting for unimodular amenable groups. Joint work with Ujan Chakraborty and Joachim Zacharias.Nikolai Prochorov
Aix-Marseille University
In the 1980’s, William Thurston obtained his celebrated characterization of rational mappings. This result laid the foundation of such a field as Thurston's theory of holomorphic maps, which has been actively developing in the last few decades. One of the most important problems in this area is the questions about characterization, which is understanding when a topological map is equivalent (in a certain dynamical sense) to a holomorphic one, and classification, which is an enumeration of all possible topological models of holomorphic maps from a given class. In my talk, I am going to focus on the characterization and classification problems for the family of critically fixed branched coverings, i.e., branched coverings of the 2-dimensional sphere S^2 with all critical points being fixed. Maps of this family can be defined by combinatorial models based on planar embedded graphs, and it provides an elegant answer to the classification problem for this family. Further, I plan to explain how to understand whether a given critically fixed branched cover is equivalent to a critically fixed rational map of the Riemann sphere and provide an algorithm of combinatorial nature that allows us to answer this question. This is a joint work with Mikhail Hlushchanka.Jonathan Fraser
University of St Andrews
Let E be a non-empty compact subset of the Riemann sphere and T be a rational map of degree at least two. The associated orbital set is defined to be the backwards orbit of E under T. I will consider under what circumstances the upper box dimension of the orbital set may be expressed in terms of the upper box dimensions of the Julia set of T and the initial set E. Our results extend previous work on inhomogeneous iterated function systems and Kleinian orbital sets to the setting of complex dynamical systems. This is joint work with my PhD student Yunlong Xu.The boundary of chaos and the boundary of positive Hausdorff dimension of survivor sets for two-branch maps of the interval
Paul Glendinning
University of Manchester
Two classical problems in bifurcation theory are the characterisation of the boundary of chaos (in the sense of positive topological entropy) of families of maps, and the boundary of positive Hausdorff dimension of survivor sets in open maps (maps with 'holes' in phase space, the survivor set is the set of points whose forward iterates never land in the hole). In one dimension these two problems are effectively equivalent, and I will give a complete description of the boundary for classes of expanding maps with two monotonic branches and either a hole or a plateau. There is a natural two-parameter family associated with these maps (the position of the hole/plateau) and I will show that there are two classes of codimension one transitions for these maps (and uncountably many codimension two points!). Much of this work is joint with Clement Hege.Dimension drop and estimating the gap
Peej Ingarfield
University of Manchester
Iterated function systems (IFS) have played a key role in the study of fractals. When considering an IFS, studying what separation properties it has gives a great amount of information about the fractal associated to the IFS. In the case that the IFS satisfies the open set condition then Hutchinsons formula gives an exact form for the Hausdorff dimension of the fractal. This leads to the natural question of what can we say about Hausdorff dimension when an IFS fails to meet the open set condition or even has overlaps. In this talk we will discuss exact overlaps and how they are conjectured to be the cause of a phenomenon called dimension drop. After this context has been established we shall compare methods to estimate the amount this dimension can drop for a class of self similar measures with overlaps.Hyperbolic correspondences
Laurent Bartholdi
CNRS Lyon
In joint work with Dima Dudko and Kevin Pilgrim, we consider certain correspondences on a Riemann surface, and show that they admit a weak form of hyperbolicity: sufficiently long loops get shorter under lifting at a fixed point and closing. This applies in particular to the correspondences induced on Moduli space studied by Thurston, and lets us deduce: for any rational map on Cp with 4 post-critical points, there is a finite invariant collection of isotopy classes of curves into which every curve is attracted under repeated lifting. I will explain the theory behind all this: an encoding of maps and correspondences by an algebraic gadget called a "biset": a set with two commuting group actions.