• Location: Room V109, Mathematics Building (Callaghan Campus) The University of Newcastle
  • Dates: 12:00 pm, Fri, 1st Nov 2019 - 4:30 pm, Fri, 1st Nov 2019
  • Schedule:

    12-1: Anthony Dooley
    1-2: Lunch
    2-3: Colin Reid
    3-3.30: Tea
    3.30-4.30: Michael Barnsley

  • Speaker: Prof Anthony Dooley, University of Technology Sydney
  • Title: Classification of non-singular systems and critical dimension
  •      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 $X$.

  • Speaker: Dr Colin Reid, CARMA, The University of Newcastle
  • Title: Piecewise powers of a minimal homeomorphism of the Cantor set
  •      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.

  • Speaker: Prof Michael Barnsley, Mathematical Sciences Institute, Australian National University
  • Title: Dynamics on Fractals
  •      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.

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