arXiv subjects

Christoph Bandt

Publications and source records attributed to Christoph Bandt.

At least 19 recordsLinked to original sources

arXiv2026

Self-similar Delone sets and Pisot numbers

We consider Delone point patterns with self-similarity. Under mild conditions, the similarity factor is a Pisot number if and only if the pattern is uniformly discrete. The classical case is a Meyer set $\Lambda$ with $\Lambda\supset \theta\Lambda$ for some $\theta>1,$ for which $\theta$ must be a Pisot number or a Salem number. When $\Lambda$ contains several similar copies of itself, the case of a Salem number drops out for $\theta<2.$ On the other hand, strictly self-similar patterns with a Pisot factor must be Meyer sets. Various examples are given.

math.DS↗

arXiv2026

Quasicrystal model sets from overlapping self-similar attractors

We give a simple computational approach to mathematical quasicrystals, combining cut-and-project methods with self-similarity. Starting with a Pisot unit $\beta$ and an iterated function system $g_k(z)=\beta z +w_k, \ k=1,...,m$ in a corresponding ring of algebraic integers, we take the attractor $A$ of the conjugate system as an acceptance window. This yields a unique cut-and-project model set $\Lambda$ in the complex plane which fulfils $\Lambda=\bigcup g_k(\Lambda) .$ We describe an algorithm which directly determines $\Lambda ,$ avoiding the difficulties with the fractal structure of $A.$ Classical constructions are based on tiles $A$ of different shape. The present study continues work on models with overlaps, as introduced by Gummelt (1996), Baake and Grimm (2013), Hejda and Pelantov\'a (2016), and Hare, Mas\'akov\'a, and V\'avra (2018). In our approach, the overlaps provide a natural decoration of $\Lambda .$ The method is illustrated with a variety of pentagonal examples.

math.MG↗

arXiv2025

Elementary fractal geometry. 6. The dynamical interior of self-similar sets

On the one hand, the dynamical interior of a self-similar set with open set condition is the complement of the dynamical boundary. On the other hand, the dynamical interior is the recurrent set of the magnification flow. For a finite type self-similar set, both boundary and interior are described by finite automata. The neighbor graph defines the boundary. The neighborhood graph, based on work by Thurston, Lalley, Ngai and Wang, defines the interior. If local views are considered up to similarity, the interior obtains a discrete manifold structure, and the magnification flow is discretized by a Markov chain. This leads to new methods for the visualization and description of finite type attractors.

math.DS↗

arXiv2024

Elementary fractal geometry. 5. Weak separation is strong separation

For self-similar sets, there are two important separation properties: the open set condition and the weak separation condition introduced by Zerner, which may be replaced by the formally stronger finite type property of Ngai and Wang. We show that any finite type self-similar set can be represented as a graph-directed construction obeying the open set condition. The proof is based on a combinatorial algorithm which performed well in computer experiments.

math.DS↗

arXiv2023

Elementary fractal geometry. 4. Automata-generated topological spaces

Finite automata were used to determine multiple addresses in number systems and to find topological properties of self-affine tiles and finite type fractals. We join these two lines of research by axiomatically defining automata which generate topological spaces. Simple examples show the potential of the concept. Spaces generated by automata are topologically self-similar. Two basic algorithms are outlined. The first one determines automata for all $k$-tuples of equivalent addresses from the automaton for double addresses. The second one constructs finite topological spaces which approximate the generated space. Finally, we discuss the realization of automata-generated spaces as self-similar sets.

math.MG↗

arXiv2023

Elementary fractal geometry. 3. Complex Pisot factors imply finite type

Self-similar sets require a separation condition to admit a nice mathematical structure. The classical open set condition (OSC) is difficult to verify. Zerner proved that there is a positive and finite Hausdorff measure for a weaker separation property which is always fulfilled for crystallographic data. Ngai and Wang gave more specific results for a finite type property (FT), and for algebraic data with a real Pisot expansion factor. We show how the algorithmic FT concept of Bandt and Mesing relates to the property of Ngai and Wang. Merits and limitations of the FT algorithm are discussed. Our main result says that FT is always true in the complex plane if the similarity mappings are given by a complex Pisot expansion factor $λ$ and algebraic integers in the number field generated by $λ.$ This extends the previous results and opens the door to huge classes of separated self-similar sets, with large complexity and an appearance of natural textures. Numerous examples are provided.

math.MG↗

arXiv2022

Two new parameters for the ordinal analysis of images

Local patterns play an important role in statistical physics as well as in image processing. Two-dimensional ordinal patterns were studied by Ribeiro et al. who determined permutation entropy and complexity in order to classify paintings and images of liquid crystals. Here we find that the 2 by 2 patterns of neighboring pixels come in three types. The statistics of these types, expressed by two parameters, contains the relevant information to describe and distinguish textures. The parameters are most stable and informative for isotropic structures.

cs.CV↗

arXiv2022

Statistics and modelling of order patterns in univariate time series

Order patterns apply well to many fields, because of minimal stationarity assumptions. Here we fix the methodology of patterns of length 3 by introducing an orthogonal system of four pattern contrasts. These contrasts are statistically independent and turn up as eigenvectors of a covariance matrix both in the independence model and the random walk model. The most important contrast is turning rate. It can be used to evaluate sleep depth directly from EEG data. The paper discusses fluctuations of permutation entropy, statistical tests, and the need of new models for noises like EEG. We show how ordinal stationary processes can be constructed without any numerical values. An order by coin-tossing is a natural example. Every partially stationary probability measure on patterns of length $m$ can be extended to a stationary measure on patterns of infinite length.

math.DS↗

arXiv2020

A reproduction rate which perfectly fits Covid-19

We present a simple technique to compare the development of the Covid-19 epidemic in different regions, based only on the time series of confirmed cases. Weekly new infections, taken for every day, are interpreted as infection potential of Covid-19. We derive a robust time-varying reproduction rate for the infection potential, including asymptomatic cases, which does not depend on death rate or testing intensity. It requires few assumptions and shows a more plausible time course than official reproduction rates in several countries.

q-bio.PE↗

arXiv2020

Transparent Covid-19 prediction

We present a very simple and transparent method to interpret time series of confirmed Covid-19 cases. Roughly speaking, the analysis of weekly new infections for each day is a tool for the definition and early detection of the turning point of the epidemic. In Italy, the growth of Covid-19 activity was overcome one week ago, Austria and Switzerland followed suit. This note emphasizes the crucial delay of two weeks between infection and listing of infection in a central database. Our estimates of the time course of Covid-19 activity and reproduction rates can reduce the information gap by several days. We show that Spain and Germany are already beyond the turning point. In general the estimated reproduction rates become undercritical a short time after the main lockdown measures, and the Covid-19 activity starts to decrease afterwards. This note gives a very optimistic outlook for all regions which have taken strict containment measures.

q-bio.PE↗

arXiv2020

Elementary fractal geometry. Networks and carpets involving irrational rotations

Self-similar sets with open set condition, the linear objects of fractal geometry, have been considered mainly for crystallographic data. Here we introduce new symmetry classes in the plane, based on rotation by irrational angles. Examples without characteristic directions, with strong connectedness and small complexity were found in a computer-assisted search. They are surprising since the rotations are given by rational matrices, and the proof of the open set condition usually requires integer data. We develop a classification of self-similar sets by symmetry class and algebraic numbers. Examples are given for various quadratic number fields. .

math.MG↗

arXiv2019

Order patterns, their variation and change points in financial time series and Brownian motion

Order patterns and permutation entropy have become useful tools for studying biomedical, geophysical or climate time series. Here we study day-to-day market data, and Brownian motion which is a good model for their order patterns. A crucial point is that for small lags (1 up to 6 days), pattern frequencies in financial data remain essentially constant. The two most important order parameters of a time series are turning rate and up-down balance. For change points in EEG brain data, turning rate is excellent while for financial data, up-down balance seems the best. The fit of Brownian motion with respect to these parameters is tested, providing a new version of a forgotten test by Bienaym'e.

q-fin.ST↗

arXiv2018

Computer geometry: Rep-tiles with a hole

A cube is an 8-rep-tile: it is the union of eight smaller copies of itself. Is there a set with a hole which has this property? The computer found an interesting and complicated solution, which then could be simplified. We discuss some problems of computer-assisted research in geometry.

math.MG↗

arXiv2018

New relatives of the Sierpinski gasket

By slight modification of the data of the Sierpinski gasket, keeping the open set condition fulfilled, we obtain self-similar sets with very dense parts, similar to fractals in nature and in random models. This is caused by a complicated structure of the open set and is revealed only under magnification. Thus the family of self-similar sets with separation condition is much richer and has higher modelling potential than usually expected. An interactive computer search for such examples and new properties for their classification are discussed.

math.DS↗

arXiv2017

Finite orbits in multivalued maps and Bernoulli convolutions

Bernoulli convolutions are certain measures on the unit interval depending on a parameter $β$ between 1 and 2. In spite of their simple definition, they are not yet well understood. We study their two-dimensional density which exists by a theorem of Solomyak. To each Bernoulli convolution, there is an interval $D$ called the overlap region, and a map which assigns two values to each point of $D$ and one value to all other points of $[0,1].$ There are two types of finite orbits of these multivalued maps which correspond to zeros and potential singularities of the density, respectively. Orbits which do not meet $D$ belong to an ordinary map called $β$-transformation and exist for all $β>1.6182.$ They were studied by Erdös, Jóo, Komornik, Sidorov, de Vries and others as points with unique addresses, and by Jordan, Shmerkin and Solomyak as points with maximal local dimension. In the two-dimensional view, these orbits form address curves related to the Milnor-Thurston itineraries in one-dimensional dynamics. The curves depend smoothly on the parameter and represent quantiles of all corresponding Bernoulli convolutions. Finite orbits which intersect $D$ have a network-like structure and can exist only at Perron parameters $β.$ Their points are intersections of extended address curves, and can have finite or countable number of addresses, as found by Sidorov. For an uncountable number of parameters, the central point $\frac12$ has only two addresses. The intersection of periodic address curves can lead to singularities of the measures. We give examples which are not Pisot or Salem parameters. It seems that all singularities of Bernoulli convolutions are related to network-like orbits. The paper is self-contained and includes many illustrations.

math.DS↗

arXiv2017

Crude EEG parameter provides sleep medicine with well-defined continuous hypnograms

To evaluate EEG data, one can count local maxima and minima on a fine scale, in a sliding window analysis. This straightforward calculation, which simplifies and improves previous work on permutation entropy, directly defines a good proxy for brain activity in an EEG channel during an epoch of 30 seconds. Different channels and persons can be compared when they are measured with the same device and prefiltering options. This could lead to a rigorously defined and suitably standardized biomarker of cortex activity, like blood pressure or laboratory values. Applied to sleep EEG, the algorithm yields hypnograms with continuous scale which show amazing coincidence with sleep stage annotation by trained experts. Although produced by a crude method, continuous hypnograms provide a lot of details. For example, sleep depth usually decreases from evening to morning even within the same annotated sleep stage, except for REM phases where mean sleep depth is rather constant, but different in frontal and parietal channels. The diagnostic potential of the method is demonstrated with two hypnograms of narcoleptic patients. In all 10 subjects, infra-slow oscillations of activity with a wavelength between 30s and two minutes were clearly seen, particularly strong at the onset of sleep and in S2 phases. The suggested method needs to be checked and improved. In its present form it seems already an appropriate tool for screening long-term EEG data.

stat.AP↗

arXiv2017

A single fractal pinwheel tile

The pinwheel triangle of Conway and Radin is a standard example for tilings with self-similarity and statistical circular symmetry. Many modifications were constructed, all based on partitions of triangles or rectangles. The fractal example of Frank and Whittaker requires 13 different types of tiles. We present an example of a single tile with fractal boundary and very simple geometric structure which has the same symmetry and spectral properties as the pinwheel triangle.

math.DS↗

arXiv2016

The two-dimensional density of Bernoulli Convolutions

Bernoulli convolutions form a one-parameter family of self-similar measures on the unit interval. We suggest to study their two-dimensional density which has an intricate combinatorial structure. Visualizing this structure we discuss results of Erdös, Jóo, Komornik, Sidorov, de Vries, Jordan, Shmerkin and Solomyak, Feng and Wang. We emphasize the rôle of finite orbits of associated multivalued maps and mention a few new properties and examples.

math.DS↗