MASCOS Workshop on Algebraic Dynamics

University of New South Wales, Sydney, Australia
February 14 - February 18, 2005

Titles, Abstracts and Selected Talks/Papers

Invited Talks:

Introduction to commutative algebraic dynamics
Klaus Schmidt

These two 45-minutes lectures will provide a brief introduction to the dynamics of commuting automorphisms of compact Abelian groups, emphasizing the concrete examples and questions that drive the subject and their connections with commutative algebra. Topics will include:

Basic References:

Entropy in algebraic dynamics
Douglas Lind

This lecture will focus on entropy for the algebraic systems introduced in Klaus's lectures. Topics covered will be:

Basic References:

Aperiodic order and Dynamical Systems I and II
Robert V. Moody, Daniel Lenz

A lattice in a real Euclidean space is the epitome of an infinite discrete system which is highly ordered. Long-range order or aperiodic order refers to the phenomenon of discrete systems of infinite extent which still have very evident order but are either partially or totally deficient in periodic symmetries. The order appears now in several ways, the most notable being the repetition of local structure (albeit aperiodically) and, most importantly, through the existence of a strong pure point component in the diffraction. Famous examples are the Penrose tilings, the Fibonacci substitution sequences, and the actual physical examples of quasi-crystalline materials.

Just as in the case of statistical mechanics, it has proven extremely useful to study aperiodic structures not just as individuals in isolation, but rather as members of some larger family of closely related aperiodic structures; for example, all those objects whose local structures are indistinguishable up to translation. Thereby arise dynamical systems, and again, just as in statistical mechanics, the spectral theory of the associated dynamical system provides a powerful method of exploring the underlying geometry of the original structure that led to it.
Two lectures will be devoted to this interesting application of dynamical systems.

The first lecture (RVM) will be devoted to long-range aperiodic order itself. Using the familiar lattices as a starting point, we will see how many of their familiar properties can be slightly relaxed so as to accommodate a whole range of new examples. In particular we will discuss the very general cut and project method of creating aperiodic point sets. In keeping with the algebraic emphasis of the conference we will show how the quaternions over the ring of real integers of the cyclotomic field of 5th roots of unity give rise to families of aperiodic structures that are beautifully ordered and strongly arithmetic in nature. Finally we will look at the mathematical definition of diffraction and prepare for the introduction of the dynamical systems which will come in the second lecture.
A gentle introduction to this subject, in the context of tilings, is Charles Radin's Miles of Tiles, AMS.

The second lecture (DL) will focus on the actual use of dynamical systems in the study of aperiodic order.
A discrete point set gives rise to a dynamical system by completing the set of its translates in a suitable topology. Two topologies will play a role: The local topology and the autocorrelation topology.
The local topology allows one to interpret certain geometric features in terms of compactness, minimality and unique ergodicity. Furthermore, pure point diffraction can be characterised by pure point dynamical spectrum. The autocorrelation topology gives a new understanding of pure point diffraction in terms of a compactness property. For cut and project sets these two topologies are related. This gives a way to characterise certain cut and project sets in terms of dynamical systems.

Dynamics of piecewise isometries, from introductory examples to current research questions
Arek Goetz

We illustrate the beauty and complexity of piecewise isometric systems by surveying known examples and stating most resistible open problems. Piecewise isometric systems are maps with singularities that act as isometries on restricted domains. These systems give rise to intriguing yet complicated structures of cells, sets that follow the same pattern of visits to continuity domains. Such structures have been successfully studied using first return maps and algebraic tools. In the absence of a general theory, the use of algebraic tools together with a computer have been central in understanding almost all geometric examples. The talk, while rigorous in nature, will be supported by multimedia and it will be accessible to a general audience.

The 3x+1 problem
Günther Wirsching

The Collatz function is defined initially on the set of positive integers, mapping an even n to n/2 and an odd n to 3n+1. Iteration of this map gives a dynamical system on the set of positive integers (or on an appropriate larger set) which is interesting because its study leads to complicated and unsolved problems. The famous 3x+1 conjecture states that any Collatz trajectory starting with a positive integer ends up in the cycle 1-4-2. This conjecture is unsolved, but there are many intermediate results relating 3x+1 dynamics to deep methods of number theory, dynamical systems, Markov processes, p-adic measure theory and quasi-Monte Carlo methods. We present an introduction into the bizarre landscape of mathematics related to 3x+1 dynamics and indicate some problems that seem to be more tractable than the 3x+1 conjecture itself.


Asymptotic spectral distributions
Friedrich Götze

We give a survey on the appearance of universal local and global spectral distributions derived from random matrix models in different contexts. In particular, we shall describe connections to random combinatorial models, random walks under order restrictions, as well as to distribution of zeros of L-functions and to non-commutative or free probability theory.

We shall give an introduction to the framework of free probability theory and describe an analytic approach to free convolutions of (spectral) measures based on recent joint work with G. Chistyakov. The classical theory of convolution of probability measures and their limit behavior by Gnedenko, Khintchin and Kolmogorov will be reviewed in comparison to analogous results in free probability theory, which in some cases exhibit surprising differences.


Eigenvalue distributions of random matrices
Peter Forrester

As surveyed in the seminar of F. Götze, the eigenvalue distributions of Gaussian random matrices and the random matrices from the classical groups play a fundamental role in the applications of random matrices. A basic question relates to the sampling from these distributions: how can it most efficiently be carried out? Rather than having to generate a random matrix of the sought type, and then computing its eigenvalues, it is now known that the characteristic polynomials in question satisfy simple recurrences with random coefficients. Thus the distributions can be sampled by computing the characteristic polynomials from the recurrences, and then computing its zeros. I'll review these developments, and explain my own contribution. One aspect of the latter (in joint work with Eric Rains) relates to the eigenvalue distribution of certain rank 1 perturbations, or equivalently the zeros of some random rational functions.


Costs of equivalence relations and group actions
Tony Dooley

Much work has been done studying amenable group actions, but until recently, it has been difficult to handle non-amenable actions. A break-through was made with work of Levitt, Kechris, Gaboriau, which defines a new invariant, the cost of a group action (or equivalence relation). Gaboriau showed how to use this invariant to distinguish between group actions of, for example, the free group on two generators and the free group on three generators.
In joint work with Golodets, we used the theory of index cocycles of Feldman, Sutherland and Zimmer to calculate the cost of equivalence relations which are finite extensions. This enables us to resolve some conjectures of Gaboriau, and also to show that many group actions cannot be isomorphic.
I will give an introduction to the theory of costs and an outline of our main results.

Dynamical systems generated by rational maps over finite fields and rings
Igor Shparlinski

We describe dynamical systems generated by iterations of rational functions over finite fields and rings. Besides their intrinsic interest, such dynamical systems have also been used as sources of reliable pseudorandom numbers in Monte Carlo methods and cryptography. We present a survey of recent developments, and outline several open problems.
More details and references can be found in:

Maps over finite fields: integrability and reversibility
Franco Vivaldi

In the theory of dynamical systems, integrability (existence of invariants of the motion) and reversibility (existence of conjugacy with inverse map) are important structural properties. We let two-dimensional algebraic mappings act on finite coordinate fields, and present experimental evidence for the existence of limit distributions of the length of the orbits for the integrable and reversible case. Such distributions feature considerable rigidity (independence from the mapping).

Contributed Talks:

Structure of totally disconnected groups via compact open subgroups; an overview of the theory and its applications.
Udo Baumgartner, University of Newcastle

It has been 10 years since George Willis introduced methods to analyze a totally disconnected, locally compact group via its action, by conjugation, on the set of its compact open subgroups. I will give an overview over the theory and its applications. These applications include areas such as random walks on groups, ergodic Zd-actions by group automorphisms and the analysis of the structure of automorphism groups of locally finite graphs and other locally finite geometric objects.

Macroscopic dynamics, almost-invariance, and almost-cyclicity
Gary Froyland, UNSW

Dynamical systems are often transitive, although this transitivity is sometimes very weak. In some situations it can be of interest to divide the phase space into large regions, between which there is relatively little communication of trajectories. We discuss fast, simple algorithms to find such divisions and state relevant convergence results. An adaptive algorithm is developed to efficiently deal with situations where the boundaries of the weakly communicating regions are complicated. We touch on the connections between ergodic-theoretic and geometric descriptions of such decompositions.

An algebraic geometric approach to integrable maps of the plane
Danesh Jogia, UNSW

We show that the dynamics of a birational map on an elliptic curve over a field is, typically, conjugate to addition by a point (under the associated group law). When the field is taken to be the function field of rational complex functions of one variable, this amounts to an algebraic geometric version of the Arnol'd-Liouville integrability theorem for planar integrable maps. When the result is applied to finite fields, it helps to explain some universal features of the periodic orbit distribution function for the reductions of integrable maps.

A geometric approach to q-Painlevé equations and their hypergeometric solutions
Kenji Kajiwara, University of Sydney

q-Painlevé equations are a class of nonlinear second order q-difference equations which are considered to be the most fundamental integrable systems. We present an elementary algebro-geometric formulation for q-Painlevé equations such that they are regarded as the non-autonomous deformation of addition formula of cubic curves on P2. By using this formulation, we construct hypergeometric solutions for q-Painlevé equations. From this result we obtain a coalescence cascade of hypergeometric functions, starting from the very-well-posed basic hypergeometric series 10W9 on the top, ending by the Airy function.

Directed graphs for higher dimensional dynamical systems
David Pask, University of Newcastle

We shall show, by example, how to associate a directed graph to certain higher dimensional shifts. The resulting directed graph may then be viewed as a higher dimensional structure in its own right and the original shift space recovered.

The entropy of the visible points
Peter Pleasants, University of Queensland

The set of visible integer points in n-space, though simple to describe, is not a Meyer set and therefore not a regular model set. In fact it has a positive density of arbitrarily large holes, so cannot be a Meyer set up to a set of measure zero, and it was the first such set to be shown to have a pure point diffraction spectrum. I shall describe how to calculate its topological entropy, which turns out to be nonzero. A similar analysis can be applied to the kth-power-free numbers in dimension 1, but requires an unproved, though widely believed, hypothesis. I shall also describe how these sets can be viewed as nonregular model sets with adelic internal space.

Pisot substitutions and (limit-) quasiperiodic model sets
Bernd Sing, University of Bielefeld

It is conjectured that all (irreducible) one-dimensional Pisot substitution sequences can be described as model sets. We will explain this conjecture and its equivalent formulations explicitly in the case where the Pisot-Vijayaraghavan number fails to be a unit. Additional to the well-studied unit case with Euclidean internal space, the internal space then also has p-adic components.

Rational trigonometry for finite fields
Norman Wildberger, UNSW

Rational trigonometry provides a rational alternative to the usual sin and cos formulation, with the very sizable advantage that it works over any field. It allows the beginnings of a metrical theory of algebraic geometry. This will be an introductory talk, with some cute pictures.

Folding Transformations of the Painlevé Equations
Nick Witte, University of Melbourne

The six Painlevé equations are Hamiltonian dynamical systems {q(t),p(t);t,H(t)} evolving with time t in C under the action of a non-autonomous Hamiltonian H(t) with the integrable character that the only movable singularities of q(t) are simple isolated poles. The symmetry groups of the Bäcklund transformations for these systems, that is those that preserve t, are the extended affine Weyl groups of A1, A2, A3, B2 and D4.
A purely algebraic theory for the Painlevé equations can then be formulated employing the fundamental reflections and Dynkin diagram automorphisms of the particular group. However it has been recently revealed that all the Painlevé equations possess additional transformations, ones that do not preserve t, almost 100 years after the first example was found in the case of the second Painlevé equation. These are the folding transformations.

last modified on 22 Feb 2005