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:
Furstenberg's example (multiplication by 2 and 3 on the
circle). Closed invariant subsets and invariant probability
measures.
Single toral automorphisms: eigenvalues and dynamical
properties (ergodicity, expansiveness, invariant measures,
invariant sets).
Commuting toral automorphisms: examples and dynamical
properties (expansiveness, invariant measures).
Commuting automorphisms of compact Abelian groups: the dual
module. Examples.
Dynamical properties: the "dictionary" between commutative
algebra and dynamics. More examples.
Directions in current research (e.g. the isomorphism problem).
Basic References:
Survey article "Symbolic and Algebraic Dynamical Systems" by
Lind and Schmidt, available
here (a chapter in the Handbook of Dynamical Systems).
Schmidt, "Dynamical Systems of Algebraic Origin", Birkhauser
(1995).
This lecture will focus on entropy for the algebraic systems
introduced in Klaus's lectures. Topics covered will be:
Basic definitions, and why all reasonable definitions of
entropy coincide in the algebraic setting.
Examples for toral and solenoidal entropy, first appearance of
p-adic ideas.
Lehmer's problem and the "smallest" algebraic integer.
Mahler measure and the computation of entropy in the principal
ideal case.
Complete description of the entropy rank one situation (e.g.
commuting toral automorphisms.
Subspace entropy for lower rank and current research
directions.
Basic References:
For general information about entropy, see Peter Walter's book
"Ergodic Theory"
For the first use of p-adic ideas, see "Automorphisms of solenoids and
p-adic entropy" (Lind and Ward), Ergodic Th. & Dyn. Sys. 8
(1988) 411-419.
For the use of Mahler measure to compute entropy, see Chapter V
of Schmidt's book mentioned above, or "Mahler measure and
entropy for automorphisms of compact groups" (Lind, Schmidt,
Ward), Inventiones Mathematicae 101 (1990) 593-629.
The entropy rank one situation is contained in "Algebraic
Z^d-actions of entropy rank one" (Einsiedler and Lind),
Trans. Amer. Math. Soc. 356 (2004) 1799-1831; available
here.
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.
References:
Jeffrey C. Lagarias:
The 3x+1 problem and its generalizations ,
Amer. Math. Monthly 92 (1985) 3-23.
This paper is available here.
J. Simons, B. de Weger: Theoretical and computational bounds for
m-cycles of the 3n+1 problem, to appear in: Acta
Arithmetica. A preprint is available
here.
Günther J. Wirsching: The Dynamical System Generated by the
3n+1 Function, Springer Lecture Notes in Mathematics 1681
(1998)
Günther J. Wirsching:
On the problem of positive predecessor density in 3n+1 dynamics, Discrete and
Continuous Dynamical Systems, 9 No. 3 (May 2003) 771-787.
A preprint is available here.
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.
References:
Mehta, M. L.: Random Matrices. Boston (1991).
Katz, N. and Sarnak, P.: Random matrices, Frobenius eigenvalues, and
monodromy. AMS, (1999).
Berry, M. V. and Keating, J. P.: The Riemann zeros and eigenvalue
asymptotics,
Siam Review (1999) 236-266
Voiculescu, D. V., Dykema, K. J. and Nica, A.: Free random
variables. AMS,
(1992)
Voiculescu, D.: Lectures on free probability theory. In: Lectures on
probability theory and statistics (Saint-Flour, 1998), Lecture Notes in Math.,
1738, Springer (2000) 279-349
Tracy, C. A. and Widom, H.: Distribution functions for largest eigenvalues
and
their applications. In Proceedings
of the International Congress of Mathematicians, Vol. I
(Beijing, 2002) 587-596
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.
References:
I. Dumitrui, A. Edelman. Matrix models for beta ensembles.
J. Math. Phys. 43 (2002) 5830-5847.
R. Killip, I. Nenciu. Matrix models for circular ensembles.
Int. Math. Res. Not. 50 (2004) 2665-2701.
P.J. Forrester and E.M. Rains. Interpretations of some parameter
dependent generalizations of classical matrix ensembles.
Prob. Theory Relat. Fields 131 (2005) 1-61.
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:
G. Everest, A. J. van der Poorten, I. E. Shparlinski and T. Ward,
Recurrence sequences, Amer. Math. Soc. (2003).
H. Niederreiter and I. E. Shparlinski,
`Dynamical systems generated by rational functions',
Lect. Notes in Comp. Sci.,
Springer-Verlag, Berlin, 2643 (2003) 6-17.
H. Niederreiter and I. E. Shparlinski,
`Recent advances in the theory of nonlinear pseudorandom number
generators', Proc. Conf. on Monte Carlo and Quasi-Monte Carlo Methods, 2000,
Springer-Verlag, Berlin (2002) 86-102.
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).
References:
J.A.G. Roberts and F. Vivaldi, Arithmetical method to detect
integrability in maps, Phys. Rev. Lett. 90 (2003) 034102.
J.A.G. Roberts, D. Jogia and F. Vivaldi, The Hasse-Weil bound and
integrability detection in rational maps, J. Nonlinear Math. Phys.
10 (2003) 165-179.
J.A.G. Roberts and F. Vivaldi, Signature of time-reversal symmetry in
polynomial automorphisms over finite fields, preprint.
D. Jogia, J. A. G. Roberts and F. Vivaldi, An algebraic geometric
approach to integrable maps of the plane, preprint.
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 Z^{d}-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
P^{2}. 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
_{10}W_{9} 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.
M. Baake, R.V. Moody and P. A. B. Pleasants:
Diffraction from visible lattice points and k-th power free integers,
Discr. Math. 221 (2000) 3-42.
This paper is available
here.
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.
The very artfully designed lecture slides can be found
here .
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 A_{1}, A_{2}, A_{3}, B_{2}
and D_{4}. 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.
The lecture slides are available
here (printer optimized pdf) or
here (postscript).