Mondays at 2pm
Frank Adams 1, Alan Turing Building
Contacts Donald Robertson Yotam Smilansky
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.Dan Nicks
University of Nottingham
Gabriel Fuhrmann
Durham University
Eng-Jon Ong
Queen Mary University of London
Rachid El Harti
Hassan University
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.