CRC-MASCOS Workshop:
Symmetries and Invariants of Dynamical Systems




Bielefeld University, Germany
24th - 26th September 2008


Program

All talks take place in the Common Room, V3-201, in the Bielefeld University main building. Click here for an description of the main building.

Tentative Schedule:

Wednesday, Sept 24:

  10:00-11:00 Gary Froyland
Morning Tea Break
11:30-12:30 Sergei Pilyugin
Lunch break
14:00-15:00 Thorsten Hüls
Afternoon Tea Break
16:00-17:00 Mirko Hessel-von Molo
Thursday, Sept 25:

  10:30-11:30 John Roberts
Lunch break
14:00-15:00 Richard Miles
Afternoon Tea Break
16:00-17:00 Gerhard Keller
Friday, Sept 26:

  10:30-11:30 Tom Ward
Lunch break
14:00-15:00 Matthias Birkner
Afternoon Tea Break
15:30-16:30 Christoph Richard


Titles and Abstracts:

Christoph Richard

Dimer models and Mahler measures

The patch counting entropy of some dimer models of statistical mechanics appears to be a Mahler measure. In particular, this is the case for domino tilings on the square lattice and for lozenge tilings on the triangular lattice. The connection to corresponding models from algebraic dynamical systems is currently not fully understood. We will give an overview of these problems, by discussing various results from the mathematical and from the physical literature.

Gary Froyland

Coherent sets and Isolated Spectrum for Perron-Frobenius Cocycles

Transport and mixing processes play an important role in many natural phenomena. Ergodic theoretic approaches to identifying slowly mixing structures in autonomous systems have been developed around the Perron-Frobenius operator and its eigenfunctions. We describe an extension of these techniques to non-autonomous systems in which one can observe {\em time dependent}, but slowly dispersive structures, which we term {\em coherent sets}. We study cocycles generated by expanding interval maps and the rates of decay for functions of bounded variation under the action of the associated Perron-Frobenius cocycles. We show that when the generators are piecewise affine and share a common Markov partition, the Lyapunov spectrum of the Perron-Frobenius cocycle has at most finitely many isolated points. We also state a strengthened version the Multiplicative Ergodic Theorem for non-invertible matrices, and develop a numerical algorithm to approximate the Oseledets subspaces that describe coherent sets. [Joint work with Simon Lloyd (UNSW) and Anthony Quas (UVic).]

John Roberts

Period distribution function for reversible rational maps over finite fields.

Reversible rational maps are those maps in d-dimensional space that can be written as the composition of 2 rational involutions. Numerical evidence indicates some universal (i.e. map independent) features when these maps are reduced over finite fields. An example is the distribution of the normalized lengths of the periodic orbits in this finite phase space (in the large field limit). We review this and other universal features and show how a combinatoric model based on composing random involutions can explain them. [This is joint work with F Vivaldi (London)].
Gerhard Keller

Globally coupled piecewise expanding maps with bistable behaviour

Among the simplest globally coupled systems of chaotic maps are systems of piecewise monotone interval maps coupled by their mean field. Around 1990, Kaneko studied globally coupled logistic maps. He observed (numerically) that the fluctuations of the mean field do not necessarily scale strength, and he coined the term "violation of the law of large numbers" for this behaviour. A bit later, Ershov and Potapov observed essentially the same phenomenon in systems of globally coupled tent maps and explained it by semi-rigorous arguments. So far there is no rigorous explanation of this phenomenon. On the other hand, Esa Järvenpää proved in 1997 for systems of analytic expanding circle maps that this phenomenon does not occur at small coupling strength, and a bit later I showed that just C^2-smoothness of the invariant densities is needed. In my talk I plan to first summarize these results in a probabilistic and work in progress (with J.-B. Bardet and R. Zweimüller) where we study a parametrized family of two-to-one uniformly expanding piecewise fractional linear maps of an interval. The maps are coupled via the parameter which is a sigmoid function of the mean field. Since the maps leave a space of probability measures invariant whose distribution functions are Herglotz functions, the Perron-Frobenius operators of the maps have some "hidden" monotonicity properties. This allows to detail the dynamics of the (nonlinear) SCPFO for a broad range of coupling strengths and to show that the operator has a bifurcation from a unique stable fixed point to a pair of stable fixed points separated by a kind of hyperbolic fixed point. For all these parameters the finite systems have a unique mixing smooth invariant density. The dynamics of the SCPFO can be related to the large deviations behaviour of the finite systems at fixed time when the system size tends to infinity. The large deviations behaviour of the invariant densities is still unknown to us.

Matthias Birkner

Diffraction of stochastic point sets: Exactly solvable examples

Stochastic point sets are considered that display a diffraction spectrum of mixed type, with special emphasis on explicitly computable cases together with a unified approach of reasonable generality. Several pairs of autocorrelation and diffraction measures are discussed which show a duality structure that may be viewed as analogues of the Poisson summation formula for lattice Dirac combs. [Joint work with M. Baake (Bielefeld) and R.V. Moody (Victoria)]

Mirko Hessel- von Molo

Transfer operators for structured systems: from equivariant systems to coupled cell systems

We consider structural properties of the transfer operator for coupled dynamical systems. In case the coupling structure is (globally) symmetric, techniques from linear representation theory can be employed to decompose the space of measures and the transfer operator acting on it. In the more general case of coupled cell systems, global symmetries need not be present even for highly structured systems. We present a new decomposition of the transfer operator that is reflecting the network structure, and relate it to a weaker, local, notion of symmetry in the coupling structure. [Joint work with M. Dellnitz (Paderborn) and P. Mehta (Urbana-Champaign).]

Richard Miles

Entropy range problems

The problem of finding the range of entropy values resulting from a given class of discrete amenable group actions is well-known. For actions of Zd on compact abelian groups, there is a direct connection with Lehmer's problem concerning Mahler measures. By finding a new entropy formula, we expose a class of actions for which the full range of entropy values occurs and for which Lehmer's problem is not an issue.
Sergei Pilyugin

Approximate trajectories that approximately preserve some dynamical structures

We consider dynamical systems with multidimensional time (for example, lattice systems generated by discretizations of PDEs). One of the problems discussed in the talk is as follows: Under which conditions, an approximate trajectory that is almost a traveling wave is close to an exact traveling wave? Some other dynamical structures are studied as well.
Thorsten Hüls

Numerical computation of dichotomy rates and projectors in discrete time

We introduce a characterization of exponential dichotomies for linear difference equations that can be tested numerically and enables the approximation of dichotomy rates and projectors with high accuracy. The test is based on computing the bounded solutions of a specific inhomogeneous difference equation. For this task a boundary value and a least squares approach is applied. The results are illustrated using H\'enon's map. We compute approximations of dichotomy rates and projectors of the variational equation, along a homoclinic orbit and an orbit on the attractor. Errors that occur, when restricting the infinite dimensional problem to a finite interval, are analyzed in detail.
Tom Ward

Directional dynamics

An overview of some directional dynamics for commuting maps, including expansive subdynamics, entropy rank, and recent work on directional orbit growth.
last modified on 18 Sep 2008