Wednesday, Sept 24:

Thursday, Sept 25:

Friday, Sept 26:

Christoph Richard  Dimer models and Mahler measuresThe 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 PerronFrobenius CocyclesTransport 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 PerronFrobenius operator and its eigenfunctions. We describe an extension of these techniques to nonautonomous 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 PerronFrobenius cocycles. We show that when the generators are piecewise affine and share a common Markov partition, the Lyapunov spectrum of the PerronFrobenius cocycle has at most finitely many isolated points. We also state a strengthened version the Multiplicative Ergodic Theorem for noninvertible 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 ddimensional 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 behaviourAmong 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 semirigorous 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^2smoothness 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 twotoone 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 PerronFrobenius 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 examplesStochastic 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 systemsWe 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 (UrbanaChampaign).] 
Richard Miles  Entropy range problemsThe problem of finding the range of entropy values resulting from a given class of discrete amenable group actions is wellknown. For actions of Z^{d} 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 structuresWe 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 timeWe 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 dynamicsAn overview of some directional dynamics for commuting maps, including expansive subdynamics, entropy rank, and recent work on directional orbit growth. 