Christoph Bandt

Entropy of interval maps via permutations

For piecewise monotone interval maps, we show that the Kolmogorov–Sinai entropy can be obtained from order statistics of the values in a generic orbit. A similar statement holds for topological entropy.

Disk-Like Self-Affine Tiles in R2

We give simple necessary and sufficient conditions for self-affine tiles in R2 to be homeomorphic to a disk.

On the open set condition for self-similar fractals

For self-similar sets, the existence of a feasible open set is a natural separation condition which expresses geometric as well as measure-theoretic properties. We give a constructive approach by defining a central open set and characterizing those points which do not belong to feasible open sets.

On the Mandelbrot set for pairs of linear maps

A Mandelbrot set for pairs of complex linear maps was introduced by Barnsley and Harrington in 1985. Bousch proved that is locally connected, and Odlyzko and Poonen studied the related set of all complex roots of polynomials with coefficients 0 and 1. We give an algorithm to construct this set and study its geometric structure. In contrast to the Mandelbrot set for quadratic maps, it is shown that is not simply connected.

A simple classification tool for single-trial analysis of ERP components.

Event-related potentials (ERPs) were recorded by measuring a dense sensor EEG from eight healthy volunteers in a visual oddball experiment. Single trials were analyzed with an extremely simple high-dimensional version of discriminant analysis. The question was how many of the target trials contribute to the average P3, and to test whether other components in the ERP are sensitive to discriminate between target and non-target trials. One common classification rule for all participants expressing the P3 component correctly classified 88% of the ERPs of all subjects in response to a target or non-target trial. For four of the eight participants, there were strong differences in an early ERP component over the occipital recording sites. Their individual classification rules, obtained from the training data in the time interval up to 200 ms, correctly classified 85% of the trials of the test data.

Fractal n-gons and their Mandelbrot sets

We consider self-similar sets in the plane for which a cyclic group acts transitively on the pieces. Examples like n-gon Sierpinski gaskets, Gosper snowflake and terdragon are well known, but we study the whole family. For each n our family is parametrized by the points in the unit disc. Due to a connectedness criterion, there are corresponding Mandelbrot sets which are used to find various new examples with interesting properties. The Mandelbrot sets for n > 2 are regular-closed, and the open set condition holds for all parameters on their boundary, which is not known for the case n = 2.

Permutation Entropy and Order Patterns in Long Time Series

While ordinal techniques are commonplace in statistics, they have been introduced to time series fairly recently by Hallin and coauthors. Permutation entropy, an average of frequencies of order patterns, was suggested by Bandt and Pompe in 2002 and used by many authors as a complexity measure in physics, medicine, engineering, and economy. Here a modified version is introduced, the “distance to white noise.” For datasets with tens of thousands or even millions of values, which are becoming standard in many fields, it is possible to study order patterns separately, determine certain differences of their frequencies, and define corresponding autocorrelation type functions. In contrast to classical autocorrelation, these functions are invariant with respect to nonlinear monotonic transformations of the data. For order three patterns, a variance-analytic “Pythagoras formula” combines the different autocorrelation functions with our new version of permutation entropy. We demonstrate the use of such correlation type functions in sliding window analysis of biomedical and environmental data.

Analysis of Bivariate Coupling by Means of Recurrence

In the analysis of coupled systems, various techniques have been developed to model and detect dependencies from observed bivariate time series. Most well-founded methods, like Granger-causality and partial coherence, are based on the theory of linear systems: on correlation functions, spectra and vector autoregressive processes. In this paper we discuss a nonlinear approach using recurrence.

Self-similar sets with an open set condition and great variety of overlaps

For a very simple family of self-similar sets with two pieces, we prove, using a technique of Solomyak, that the intersection of the pieces can be a Cantor set with any dimension in [0, 0.2] as well as a finite set of any cardinality 2 m . The main point is that the open set condition is fulfilled for all these examples.

THE GEOMETRY OF A PARAMETER SPACE OF INTERACTING PARTICLE SYSTEMS

We study a four-parameter family of interacting particle systems containing the basic voter model and contact processes. Two processes in this family are related by duality or thinning if and only if their parameters belong to the same orbit of a certain one-dimensional group of linear mappings. This shows that many duals exist.