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Sabrina Kombrink

Publications and source records attributed to Sabrina Kombrink.

11 recordsLinked to original sources
arXiv2026

On the geometry of generalised Koch snowflakes

We consider the geometry of a class of fractal sets in $\mathbb{R}^{2}$ that generalise the famous Koch curve and Koch snowflake. While the classical Koch curve is defined by an iterative process that divides a line segment into three parts and replaces the middle part by the legs of an isosceles triangle 'above' the line segment, in this more general setting, a choice can be made at each iteration as to whether to place this triangle 'above' or 'below' the line segment. The resulting fractals bear a striking visual resemblance to curves appearing in nature, such as coastlines and snowflakes. While these fractals can be generated by a random process that flips a coin each time to decide the orientation of the triangle, leading to 'almost sure' results for their geometrical properties, we define and study them deterministically to provide exact results. In particular, we show, using the theory of non-integer expansions, that the set of all possible values for the area enclosed by these generalised Koch curves is a closed interval. Moreover, we prove that the union of all these generalised snowflakes does not contain an open set, and has zero $2$-dimensional Lebesgue measure. Complementing these results, using arguments from calculus and fractal geometry, namely properties of geometric series and Frostman's Lemma, we show that each generalised Koch curve has infinite length and the same Hausdorff dimension as its classical counterpart. Further, we also give a classification for when a generalised Koch curve is a quasicircle.

math.DS↗
arXiv2023

On bounds for the remainder term of counting functions of the Neumann Laplacian on domains with fractal boundary

We provide a new constructive method for obtaining explicit remainder estimates of eigenvalue counting functions of Neumann Laplacians on domains with fractal boundary. This is done by establishing estimates for first non-trivial eigenvalues through Rayleigh quotients. A main focus lies on domains whose boundary can locally be represented as a limit set of an IFS, with the classic Koch snowflake and certain Rohde snowflakes being prototypical examples, to which the new method is applied. Central to our approach is the construction of a novel foliation of the domain near its boundary.

math.SP↗
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

Renewal theorems for a class of processes with dependent interarrival times and applications in geometry

Renewal theorems are developed for point processes with interarrival times $W_n=ξ(X_{n+1}X_n\cdots)$, where $(X_n)_{n\in\mathbb Z}$ is a stochastic process with finite state space $Σ$ and $ξ\colonΣ_A\to\mathbb R$ is a Hölder continuous function on a subset $Σ_A\subsetΣ^{\mathbb N}$. The theorems developed here unify and generalise the key renewal theorem for discrete measures and Lalley's renewal theorem for counting measures in symbolic dynamics. Moreover, they capture aspects of Markov renewal theory. The new renewal theorems allow for direct applications to problems in fractal and hyperbolic geometry; for instance, results on the Minkowski measurability of self-conformal sets are deduced. Indeed, these geometric problems motivated the development of the renewal theorems.

math.PR↗
arXiv2017

Minkowski measurability of infinite conformal graph directed systems and application to Apollonian packings

We give conditions for the existence of the Minkowski content of limit sets stemming from infinite conformal graph directed systems. As an application we obtain Minkowski measurability of Apollonian gaskets, provide explicit formulae of the Minkowski content, and prove the analytic dependence on the initial circles. Further, we are able to link the fractal Euler characteristic, as well as the Minkowski content, of Apollonian gaskets with the asymptotic behaviour of the circle counting function studied by Kontorovich and Oh. These results lead to a new interpretation and an alternative formula for the Apollonian constant. We estimate a first lower bound for the Apollonian constant, namely $0{.}055$, partially answering an open problem by Oh of 2013. In the higher dimensional setting of collections of disjoint balls, generated e.\,g.\ by Kleinian groups of Schottky type, we prove that all fractal curvature measures exist and are constant multiples of each other. Further number theoretical applications connected to the Gauss map and to the Riemann $ζ$-function illustrate our results.

math.DS↗
arXiv2016

A complex Ruelle-Perron-Frobenius theorem for infinite Markov shifts with applications to renewal theory

We prove a complex Ruelle-Perron-Frobenius theorem for Markov shifts over an infinite alphabet, whence extending results by M. Pollicott from the finite to the infinite alphabet setting. As an application we obtain an extension of renewal theory in symbolic dynamics, as developed by S. P. Lalley and in the sequel generalised by the second author, now covering the infinite alphabet case.

math.DS↗
arXiv2015

Minkowski content and fractal Euler characteristic for conformal graph directed systems

We study the (local) Minkowski content and the (local) fractal Euler characteristic of limit sets $F\subset\mathbb R$ of conformal graph directed systems (cGDS) $Φ$. For the local quantities we prove that the logarithmic Cesàro averages always exist and are constant multiples of the $δ$-conformal measure. If $Φ$ is non-lattice, then also the non-average local quantities exist and coincide with their respective average versions. When the conformal contractions of $Φ$ are analytic, the local versions exist if and only if $Φ$ is non-lattice. For the non-local quantities the above results in particular imply that limit sets of Fuchsian groups of Schottky type are Minkowski measurable, proving a conjecture of Lapidus from 1993. Further, when the contractions of the cGDS are similarities, we obtain that the Minkowski content and the fractal Euler characteristic of $F$ exist if and only if $Φ$ is non-lattice, generalising earlier results by Falconer, Gatzouras, Lapidus and van Frankenhuijsen for non-degenerate self-similar subsets of $\mathbb R$ that satisfy the open set condition.

math.DS↗
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↗
arXiv2011

Minkowski Content and local Minkowski Content for a class of self-conformal sets

We investigate (local) Minkowski measurability of $\mathcal C^{1+α}$ images of self-similar sets. We show that (local) Minkowski measurability of a self-similar set $K$ implies (local) Minkowski measurability of its image $F$ and provide an explicit formula for the (local) Minkowski content of $F$ in this case. A counterexample is presented which shows that the converse is not necessarily true. That is, $F$ can be Minkowski measurable although $K$ is not. However, we obtain that an average version of the (local) Minkowski content of both $K$ and $F$ always exists and also provide an explicit formula for the relation between the (local) average Minkowski contents of $K$ and $F$.

math.DS↗
arXiv2011

Fractal curvature measures and Minkowski content for one-dimensional self-conformal sets

We show that the fractal curvature measures of invariant sets of one-dimensional conformal iterated function systems satisfying the open set condition exist, if and only if the associated geometric potential function is nonlattice. Moreover, in the nonlattice situation we obtain that the Minkowski content exists and prove that the fractal curvature measures are constant multiples of the $δ$-conformal measure, where $δ$ denotes the Minkowski dimension of the invariant set. For the first fractal curvature measure, this constant factor coincides with the Minkowski content of the invariant set. In the lattice situation we give sufficient conditions for the Minkowski content of the invariant set to exist, contrasting the fact that the Minkowski content of a self-similar lattice fractal never exists. However, every self-similar set satisfying the open set condition exhibits a Minkowski measurable $\mathcal{C}^{1+α}$ diffeomorphic image. Both in the lattice and nonlattice situation average versions of the fractal curvature measures are shown to always exist.

math.MG↗