A non-singular measurable dynamical system is a measure space $X$ whose measure $\mu$ has the property that $\mu $ and $\mu \circ T$ are equivalent measures (in the sense that they have the same sets of measure zero). Here $T$ is a bimeasurable invertible transformation of $X$. The basic building blocks are the \emph{ergodic} measures. Von Neumann proposed a classification of non-singular ergodic dynamical systems, and this has been elaborated subsequently by Krieger, Connes and others. This work has deep connections with C*-algebras. I will describe some work of myself, collaborators and students which explore the classification of dynamical systems from the point of view of measure theory. In particular, we have recently been exploring the notion of critical dimension, a study of the rate of growth of sums of Radon-Nikodym derivatives $\Sigma_{k=1}^n \frac{d\mu \circ T^k}{d\mu}$. Recently, we have been replacing the single transformation $T$ with a group acting on the space $X$.

Let $X$ be the Cantor set and let $g$ be a minimal homeomorphism of $X$ (that is, every orbit is dense). Then the topological full group $\tau[g]$ of $g$ consists of all homeomorphisms $h$ of $X$ that act 'piecewise' as powers of $g$, in other words, $X$ can be partitioned into finitely many clopen pieces $X_1,...,X_n$ such that for each $i$, $h$ acts on $X_i$ as a constant power of $g$. Such groups have attracted considerable interest in dynamical systems and group theory, for instance they characterize the homeomorphism up to flip conjugacy (Giordano--Putnam--Skau) and they provided the first known examples of infinite finitely generated simple amenable groups (Juschenko--Monod). My talk is motivated by the following question: given $h\in\tau[g]$ for some minimal homeomorphism $g$, what can the closures of orbits of $h$ look like? Certainly $h\in\tau[g]$ is not minimal in general, but it turns out to be quite close to being minimal, in the following sense: there is a decomposition of $X$ into finitely many clopen invariant pieces, such that on each piece $h$ acts a homeomorphism that is either minimal or of finite order. Moreover, on each of the minimal parts of $h$, then either $h$ or $h^{-1}$ has a 'positive drift' with respect to the orbits of $g$; in fact, it can be written in a canonical way as a conjugate of a product of induced transformations (aka first return maps) of $g$. No background knowledge of topological full groups is required; I will introduce all the necessary concepts in the talk.

I will outline a new theory of fractal tilings. The approach uses graph iterated function systems (IFS) and centers on underlying symbolic shift spaces. These provide a zero dimensional representation of the intricate relationship between shift dynamics on fractals and renormalization dynamics on spaces of tilings. The ideas I will describe unify, simplify, and substantially extend key concepts in foundational papers by Solomyak, Anderson and Putnam, and others. In effect, IFS theory on the one hand, and self-similar tiling theory on the other, are unified.
The work presented is largely new and has not yet been submitted for publication. It is joint work with Andrew Vince (UFL) and Louisa Barnsley. The presentation will include links to detailed notes. The figures illustrate 2d fractal tilings.
By way of recommended background reading I mention the following awardwinning paper: M. F. Barnsley, A. Vince, Self-similar polygonal tilings, Amer. Math. Monthly 124 (1017) 905-921.

Given a profinite group $G$, we can consider the semigroup $\mathrm{End}(G)$ of continuous homomorphisms from $G$ to itself. In general $\lambda \in\mathrm{End}(G)$ can be injective but not surjective, or vice versa: consider for instance the case when $G$ is the group $F_p[[t]$ of formal power series over a finite field, $n$ is an integer, and $\lambda_n$ is the continuous endomorphism that sends $t^k$ to $t^{k+n}$ if $k+n \ge 0$ and $0$ otherwise. However, when $G$ has only finitely many open subgroups of each index (for instance, if $G$ is finitely generated), the structure of endomorphisms is much more restricted: given $\lambda \in\mathrm{End}(G)$, then $G$ can be written as a semidirect product $N \rtimes H$ of closed subgroups, where $\lambda$ acts as an automorphism on $H$ and a contracting endomorphism on $N$. When $\lambda$ is open and injective, the structure of $N$ can be restricted further using results of Glöckner and Willis (including the very recent progress that George told us about a few weeks ago). This puts some restrictions on the profinite groups that can appear as a '$V_+$' group for an automorphism of a t.d.l.c. group.

Totally disconnected, locally compact (t.d.l.c.) groups are a large class of topological groups that arise from a few different sources, for instance as automorphism groups of a range of algebraic and combinatorial structures, or from the study of isomorphisms between finite index subgroups of a given group. A general theory has begun to emerge in recent years, based on the interaction between small-scale and large-scale structure in t.d.l.c. groups. I will give a survey of some ways in which these groups arise and some of the tools that have been developed for understanding them.

Colin Reid will present some thoughts on limits of contraction groups.

I will give an extended version of my talk at the AustMS meeting about some ongoing work with Pierre-Emmanuel Caprace and George Willis.

Given a locally compact topological group G, the connected component of the identity is a closed normal subgroup G_0 and the quotient group is totally disconnected. Connected locally compact groups can be approximated by Lie groups, and as such are relatively well-understood. By contrast, totally disconnected locally compact (t.d.l.c.) groups are a more difficult class of objects to understand. Unlike in the connected case, it is probably hopeless to classify the simple t.d.l.c. groups, because this would include for instance all simple groups (equipped with the discrete topology). Even classifying the finitely generated simple groups is widely regarded as impossible. However, we can prove some general results about broad classes of (topologically) simple t.d.l.c. groups that have a compact generating set.

Given a non-discrete t.d.l.c. group, there is always an open compact subgroup. Compact totally disconnected groups are residually finite, so have many normal subgroups. Our approach is to analyse a t.d.l.c. group G (which may itself be simple) via normal subgroups of open compact subgroups. From these we obtain lattices and Cantor sets on which G acts, and we can use properties of these actions to demonstrate properties of G. For instance, we have made some progress on the question of whether a compactly generated topologically simple t.d.l.c. group is abstractly simple, and found some necessary conditions for G to be amenable.