Workshop on Dynamical Systems and Symbolic Dynamics




Bielefeld University, Germany
17th - 19th May 2010


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.

Preliminary Program:

Monday, May 17:

  14:00 Opening
14:15-15:15 Aernout van Enter

Gibbs states versus ground states

Coffee Break
16:00-17:00 Klaus Schmidt

Sandpiles and the Harmonic Model

Tuesday, May 18:

  09:30-10:30 Tomas Persson

Transversality and beta-transformations

Coffee break
11:00-12:30 Shigeki Akiyama

Determining pure discreteness of self-affine tiling dynamical systems
(joint work with J.Y. Lee)

Lunch Break
14:00-15:00 Uwe Grimm

Entropy and letter frequencies of power-free words

Coffee Break
15:30-16:30 Franz Gähler

MLD Relations of Pisot Substitution Tilings


Wednesday, May 19:

  09:30-10:30 Jean-Paul Allouche

Inconstancy of discrete curves and sequences
(joint work of J.-P. Allouche and L. Maillard-Teyssier)

Coffee break
11:00-12:30 Tom Ward

Uniform uniform distribution of periodic points

Lunch Break


Titles and Abstracts:
Shigeki Akiyama

Determining pure discreteness of self-affine tiling dynamical systems (joint work with J.Y. Lee)

We discuss a practical algorithm to determine whether a given tiling dynamical system is pure discrete or not. Solomyak gave the overlap algorithm for this, but it was difficult to execute in practice. Switching to multi-color Delone sets, we turn it into a simple one that is applicable to all self-affine tilings once we know the tiling data: expanding matrix and digits. In the crucial part of the justification of this algorithm, we have to show a version of a folklore conjecture on the boundary of fractal tiles.

Jean-Paul Allouche

Inconstancy of discrete curves and sequences (joint work of J.-P. Allouche and L. Maillard-Teyssier)

We recall an old result due to Cauchy and to Crofton on the average number of intersections of a random line and a given curve, and we show how to deduce a measure of ``complexity'' (or of ``fluctuation'') of a discrete curve or of a sequence of real numbers, that we call ``Inconstancy''. We study the inconstancy of classical sequences. We discuss possible applications to biology, stockmarket, music...

Franz Gähler

MLD Relations of Pisot Substitution Tilings

We consider 1-dimensional, unimodular Pisot substitution tilings with three intervals, and discuss conditions under which pairs of such tilings are locally isomorhphic (LI), or mutually locally derivable (MLD). For this purpose, we regard the substitutions as automorphisms of the underlying free group. Then, if two substitutions are conjugated by an inner automorphism, the two tilings are LI, and a conjugating outer automorphism between two substitutions can often be used to prove that the two tilings are MLD. We present several examples illustrating the different phenomena arising in this context. In particular, we show how two substitution tilings can be MLD even if their substitution matrices are not equal, but only conjugate in GL(n,Z). We also illustrate how the (fractal) windows of MLD tilings can be reconstructed from each other, and discuss how the conjugating group automorphism affects the substitution generating the window boundaries.

Uwe Grimm

Entropy and letter frequencies of power-free words

We review the recent progress in the investigation of powerfree words, with particular emphasis on binary cubefree and ternary squarefree words. Besides various bounds on the entropy, we provide bounds on letter frequencies and consider their empirical distribution.

Tomas Persson

Transversality and beta-transformations

By making use of a transversality technique by Solomyak, one can analyse the Hausdorff dimension and Lebesgue measure of the attractor of some transformations similar to the baker's transformation, to get results for almost all parameters with contraction less than about 0.64. I will talk about how to extend such results to contractions larger than 0.64, for some skew-products with beta-transformations. I will also make use of some results on so-called derivable sequences, and their relation to natural extensions of beta-shifts.

Klaus Schmidt

Sandpiles and the Harmonic Model

The d-dimensional abelian sandpile model is a lattice model introduced in 1987 by Bak, Tang and Wiesenfeld as an example of `self-organized criticality'. Although this deceptively simple model has been studied quite intensively both in the physics and mathematics literature, some very basic questions about it are still open, like its properties under two different kinds of dynamics: `addition' (of grains of sand), and the shift-action. By extending an algebraic construction originally introduced by A. Vershik for Markov partitions of hyperbolic automorphisms of the 2-torus one can show that the sandpile model is closely related to a certain Zd -action by automorphisms of a compact abelian group, the `harmonic model'. In this lecture (which is based on joint work with Evgeny Verbitskiy) I'll explain this construction and its generalizations, and the conclusions that can be drawn from the connection between these systems.

Aernout van Enter

Gibbs states versus ground states

Usually one expects low temperature states to approximate ground states , which are mathematically easier than positive-temperature Gibbs states to treat (even though ground states are experimentally inaccessible). I discuss some examples in which low-temperature states fail to converge at decreasing temperatures. Moreover, I discuss some spectral aspects of Gibbs and ground states, and to what extent the distinction between dynamical and diffraction spectra might be of relevance for them.

Tom Ward

Uniform uniform distribution of periodic points

It is easy to see that (for example) the points of period k for an ergodic toral automorphism become uniformly distributed as k goes to infinity. For a mixing Z d action by commuting toral automorphisms, this gives a certain rate of uniform distribution in any chosen direction. The property of "uniform uniform" distribution is a uniformity in this rate across different directions, and it rests on estimates from Diophantine analysis. This is joint work with Richard Miles, available in preprint form at http://arxiv.org/abs/1001.0390


last modified on 12 May 2010