arXiv subjects

Steffen Winter

Publications and source records attributed to Steffen Winter.

At least 19 recordsLinked to original sources

arXiv2025

Support measures in fractal geometry

We introduce two novel families of geometric functionals-basic contents and support contents-for investigating the fractal properties of compact subsets in Euclidean space. These functionals are derived from the support measures arising in connection with the general Steiner formula due to Hug, Last, and Weil, and offer new tools for extracting geometric information beyond classical fractal dimensions. The basic contents are constructed from the support measures of the set itself, while the support contents arise from those of its parallel sets. Associated scaling exponents characterize the asymptotic behavior of these measures as the resolution parameter tends to zero. We establish a fundamental connection between the maximum of the basic scaling exponents and the outer Minkowski dimension. The proof relies on the novel notion of outer box dimension. Furthermore, we explore how support contents supply aggregated information on geometric features that is provided separately by the basic contents, and how they relate to fractal curvatures and complex dimensions. Our results provide a deeper understanding of which geometric aspects actually contribute to fractal invariants such as the (outer) Minkowski content. We provide some illustrative examples.

math.MG↗

arXiv2025

Mean Minkowski content and mean fractal curvatures of random self-similar code tree fractals

We consider a class of random self-similar fractals based on code trees which includes random recursive, homogeneous and V-variable fractals and many more. For such random fractals we consider mean values of the Lipschitz-Killing curvatures of their parallel sets for small parallel radii. Under the uniform strong open set condition and some further geometric assumptions we show that rescaled limits of these mean values exist as the parallel radius tends to 0. Moreover, integral representations are derived for these limits which recover and extend those known in the deterministic case and certain random cases. Results on the mean Minkowski content are included as a special case and shown to hold under weaker geometric assumptions.

math.PR↗

arXiv2025

Fractal curvatures and short-time asymptotics of heat content

The aim of our paper is twofold. First, we present new mathematical developments on the analysis of de Gennes' hypothesis on the short-time asymptotics of the heat content for bounded domains with smooth boundary and with fractal boundary. Second, we discuss new findings and concepts related to fractal curvatures for domains with fractal boundary. We conjecture that fractal curvatures and their scaling exponents will emerge in the short-time heat content asymptotics of domains with fractal boundary and the results discussed here are small initial contributions towards a resolution.

math.AP↗

arXiv2023

On the radial growth of ballistic aggregation and other aggregation models

For a class of aggregation models on the integer lattice $\mathbb{Z}^d$, $d\geq 2$, in which clusters are formed by particles arriving one after the other and sticking irreversibly where they first hit the cluster, including the classical model of diffusion-limited aggregation (DLA), we study the growth of the clusters. We observe that a method of Kesten used to obtain an almost sure upper bound on the radial growth in the DLA model generalizes to a large class of such models. We use it in particular to prove such a bound for the so-called ballistic model, in which the arriving particles travel along straight lines. Our bound implies that the fractal dimension of ballistic aggregation clusters in $\mathbb{Z}^2$ is 2, which proves a long standing conjecture in the physics literature.

math.PR↗

arXiv2023

On volume and surface area of parallel sets. II. Surface measures and (non-)differentiability of the volume

We prove that at differentiability points $r_0>0$ of the volume function of a compact set $A\subset {\mathbb R}^d$ (associating to $r$ the volume of the $r$-parallel set of $A$), the surface area measures of $r$-parallel sets of $A$ converge weakly to the surface area measure of the $r_0$-parallel set as $r\to r_0$. We further study the question which sets of parallel radii can occur as sets of non-differentiability points of the volume function of some compact set. We provide a full characterization for dimensions $d=1$ and $2$.

math.MG↗

arXiv2022

Mean Lipschitz-Killing curvatures for homogeneous random fractals

Homogeneous random fractals form a probabilistic extension of self-similar sets with more dependencies than in random recursive constructions. For such random fractals we consider mean values of the Lipschitz-Killing curvatures of their parallel sets for small parallel radii. Under the Uniform Strong Open Set Condition and some further geometric assumptions we show that rescaled limits of these mean values exist as the parallel radius tends to zero. Moreover, integral representations are derived for these limits which extend those known in the deterministic case.

math.PR↗

arXiv2022

Almost sure convergence and second moments of geometric functionals of fractal percolation

We determine almost sure limits of rescaled intrinsic volumes of the construction steps of fractal percolation in $\mathbb{R}^d$ for any dimension $d\geq 1$. We observe a factorization of these limit variables which allows, in particular, to determine their expectations and covariance structure. We also show convergence of rescaled expectations and variances of the intrinsic volumes of the construction steps to expectations and variances of the limit variables and give rates for this convergence in some cases. These results significantly extend our previous work that addressed only limits of expectations of intrinsic volumes.

math.PR↗

arXiv2021

Quantification of fracture roughness by change probabilities and Hurst exponents

The objective of the current study is to utilize an innovative method called 'change probabilities' for describing fracture roughness. In order to detect and visualize anisotropy of rock joint surfaces, the roughness of one-dimensional profiles taken in different directions is quantified. The central quantifiers, 'change probabilities', are based on counting monotone changes in discretizations of a profile. These probabilities, which are usually varying with the scale, can be reinterpreted as scale-dependent Hurst exponents. For a large class of Gaussian stochastic processes change probabilities are shown to be directly related to the classical Hurst exponent, which generalizes a relationship known for fractional Brownian motion. While being related to this classical roughness measure, the proposed method is more generally applicable, increasing therefore the flexibility of modeling and investigating surface profiles. In particular, it allows a quick and efficient visualization and detection of roughness anisotropy and scale dependence of roughness.

math.ST↗

arXiv2020

Characterizing digital microstructures by the Minkowski-based quadratic normal tensor

For material modeling of microstructured media, an accurate characterization of the underlying microstructure is indispensable. Mathematically speaking, the overall goal of microstructure characterization is to find simple functionals which describe the geometric shape as well as the composition of the microstructures under consideration, and enable distinguishing microstructures with distinct effective material behavior. For this purpose, we propose using Minkowski tensors, in general, and the quadratic normal tensor, in particular, and introduce a computational algorithm applicable to voxel-based microstructure representations. Rooted in the mathematical field of integral geometry, Minkowski tensors associate a tensor to rather general geometric shapes, which make them suitable for a wide range of microstructured material classes. Furthermore, they satisfy additivity and continuity properties, which makes them suitable and robust for large-scale applications. We present a modular algorithm for computing the quadratic normal tensor of digital microstructures. We demonstrate multigrid convergence for selected numerical examples and apply our approach to a variety of microstructures. Strikingly, the presented algorithm remains unaffected by inaccurate computation of the interface area. The quadratic normal tensor may be used for engineering purposes, such as mean-field homogenization or as target value for generating synthetic microstructures.

cs.CE↗

arXiv2018

Geometric functionals of fractal percolation

Fractal percolation exhibits a dramatic topological phase transition, changing abruptly from a dust-like set to a system spanning cluster. The transition points are unknown and difficult to estimate. In many classical percolation models the percolation thresholds have been approximated well using additive geometric functionals, known as intrinsic volumes. Motivated by the question whether a similar approach is possible for fractal models, we introduce corresponding geometric functionals for the fractal percolation process $F$. They arise as limits of expected functionals of finite approximations of $F$. We establish the existence of these limit functionals and obtain explicit formulas for them as well as for their finite approximations.

math.PR↗

arXiv2018

Lattice self-similar sets on the real line are not Minkowski measurable

We show that any nontrivial self-similar subset of the real line that is invariant under a lattice iterated function system (IFS) satisfying the open set condition (OSC) is not Minkowski measurable. So far, this was only known for special classes of such sets. Thereby, we provide the last puzzle-piece in proving that under OSC a nontrivial self-similar subset of the real line is Minkowski measurable iff it is invariant under a nonlattice IFS, a 25-year-old conjecture.

math.DS↗

arXiv2017

Regularly Varying Functions, Generalized contents, and the spectrum of fractal strings

We revisit the problem of characterizing the eigenvalue distribution of the Dirichlet-Laplacian on bounded open sets $Ω\subset\mathbb{R}$ with fractal boundaries. It is well-known from the results of Lapidus and Pomerance \cite{LapPo1} that the asymptotic second term of the eigenvalue counting function can be described in terms of the Minkowski content of the boundary of $Ω$ provided it exists. He and Lapidus \cite{HeLap2} discussed a remarkable extension of this characterization to sets $Ω$ with boundaries that are not necessarily Minkowski measurable. They employed so-called generalized Minkowski contents given in terms of gauge functions more general than the usual power functions. The class of valid gauge functions in their theory is characterized by some technical conditions, the geometric meaning and necessity of which is not obvious. Therefore, it is not completely clear how general the approach is and which sets $Ω$ are covered. Here we revisit these results and put them in the context of regularly varying functions. Using Karamata theory, it is possible to get rid of most of the technical conditions and simplify the proofs given by He and Lapidus, revealing thus even more of the beauty of their results. Further simplifications arise from characterization results for Minkowski contents obtained in \cite{RW13}. We hope our new point of view on these spectral problems will initiate some further investigations of this beautiful theory.

math.MG↗

arXiv2016

Localization results for Minkowski contents

It was shown recently that the Minkowski content of a bounded set $A$ in $\mathbb{R}^d$ with volume zero can be characterized in terms of the asymptotic behaviour of the boundary surface area of its parallel sets $A_r$ as the parallel radius $r$ tends to $0$. Here we discuss localizations of such results. The asymptotic behaviour of the local parallel volume of $A$ relative to a suitable second set $Ω$ can be understood in terms of the suitably defined local surface area relative to $Ω$. Also a measure version of this relation is shown: Viewing the Minkowski content as a locally determined measure, this measure can be obtained as a weak limit of suitably rescaled surface measures of close parallel sets. Such measure relations had been observed before for self-similar sets and some self-conformal sets in $\mathbb{R}^d$. They are now established for arbitrary closed sets, including even the case of unbounded sets. The results are based on a localization of Stachó's famous formula relating the boundary surface area of $A_r$ to the derivative of the volume function at $r$.

math.MG↗

arXiv2015

Lattice-type self-similar sets with pluriphase generators fail to be Minkowski measurable

A long-standing conjecture of Lapidus claims that under certain conditions, self-similar fractal sets fail to be Minkowski measurable if and only if they are of lattice type. The theorem was established for fractal subsets of $\mathbb{R}$ by Falconer, Lapidus and v.~Frankenhuijsen, and the forward direction was shown for fractal subsets of $\mathbb{R}^d$, $d \geq 2$, by Gatzouras. Since then, much effort has been made to prove the converse. In this paper, we prove a partial converse by means of renewal theory. Our proof allows us to recover several previous results in this regard, but is much shorter and extends to a more general setting; several technical conditions appearing in previous versions of this result have now been removed.

math.PR↗

arXiv2014

Estimation of fractal dimension and fractal curvatures from digital images

Most of the known methods for estimating the fractal dimension of fractal sets are based on the evaluation of a single geometric characteristic, e.g. the volume of its parallel sets. We propose a method involving the evaluation of several geometric characteristics, namely all the intrinsic volumes (i.e.\ volume, surface area, Euler characteristic etc.) of the parallel sets of a fractal. Motivated by recent results on their limiting behaviour, we use these functionals to estimate the fractal dimension of sets from digital images. Simultaneously, we also obtain estimates of the fractal curvatures of these sets, some fractal counterpart of intrinsic volumes, allowing a finer classification of fractal sets than by means of fractal dimension only. We show the consistency of our estimators and test them on some digital images of self-similar sets.

math.MG↗

arXiv2014

Minkowski content and fractal curvatures of self-similar tilings and generator formulas for self-similar sets

We study Minkowski contents and fractal curvatures of arbitrary self-similar tilings (constructed on a feasible open set of an IFS) and the general relations to the corresponding functionals for self-similar sets. In particular, we characterize the situation, when these functionals coincide. In this case, the Minkowski content and the fractal curvatures of a self-similar set can be expressed completely in terms of the volume function or curvature data, respectively, of the generator of the tiling. In special cases such formulas have been obtained recently using tube formulas and complex dimensions or as a corollary to results on self-conformal sets. Our approach based on the classical Renewal Theorem is simpler and works for a much larger class of self-similar sets and tilings. In fact, generator type formulas are obtained for essentially all self-similar sets, when suitable volume functions (and curvature functions, respectively) related to the generator are used. We also strengthen known results on the Minkowski measurability of self-similar sets, in particular on the question of non-measurability in the lattice case.

math.MG↗

arXiv2014

Scaling exponents of curvature measures

Fractal curvatures of a subset F of R^d are roughly defined as suitably rescaled limits of the total curvatures of its parallel sets F_e as e tends to 0 and have been studied in the last years in particular for self-similar and self-conformal sets. This previous work was focussed on establishing the existence of (averaged) fractal curvatures and related fractal curvature measures in the generic case when the k-th curvature measure C_k(F_e,.) scales like e^(k-D), where D ist the Minkowski dimension of F. In the present paper we study the nongeneric situation when the scaling exponents do not coincide with the dimension. We demonstrate that the possibilities for nongeneric behaviour are rather limited and introduce the notion of local flatness, which allows a geometric characterization of nongenericy in R and R^2. We expect local flatness to be characteristic also in higher dimensions. The results enlighten the geometric meaning of the scaling exponents.

math.MG↗

arXiv2012

Minkowski measurability results for self-similar tilings and fractals with monophase generators

In a previous paper [arXiv:1006.3807], the authors obtained tube formulas for certain fractals under rather general conditions. Based on these formulas, we give here a characterization of Minkowski measurability of a certain class of self-similar tilings and self-similar sets. Under appropriate hypotheses, self-similar tilings with simple generators (more precisely, monophase generators) are shown to be Minkowski measurable if and only if the associated scaling zeta function is of nonlattice type. Under a natural geometric condition on the tiling, the result is transferred to the associated self-similar set (i.e., the fractal itself). Also, the latter is shown to be Minkowski measurable if and only if the associated scaling zeta function is of nonlattice type.

math.MG↗