arXiv subjects

Marc Kesseböhmer

Publications and source records attributed to Marc Kesseböhmer.

At least 37 records · Page 2Linked to original sources

arXiv2019

Mean convergence for intermediately trimmed Birkhoff sums of observables with regularly varying tails

On a measure theoretical dynamical system with spectral gap property we consider non-integrable observables with regularly varying tails and fulfilling a mild mixing condition. We show that the normed trimmed sum process of these observables then converges in mean. This result is new also for the special case of i.i.d. random variables and contrasts the general case where mean convergence might fail even though a strong law of large numbers holds. To illuminate the required mixing condition we give an explicit example of a dynamical system fulfilling a spectral gap property and an observable with regularly varying tails but without the assumed mixing condition such that mean convergence fails.

math.DS↗

arXiv2019

Dynamics and topological entropy of 1D Greenberg-Hastings cellular automata

In this paper we analyse the non-wandering set of 1D-Greenberg-Hastings cellular automata models for excitable media with $e\geqslant 1$ excited and $r\geqslant 1$ refractory states and determine its (strictly positive) topological entropy. We show that it results from a Devaney-chaotic closed invariant subset of the non-wandering set that consists of colliding and annihilating travelling waves, which is conjugate to a skew-product dynamical system of coupled shift-dynamics. Moreover, we determine the remaining part of the non-wandering set explicitly as a Markov system with strictly less topological entropy that also scales differently for large $e,r$.

math.DS↗

arXiv2019

Measure-geometric Laplacians on the real line

Motivated by the fundamental theorem of calculus, and based on the works of Feller as well as Kac and Kre\uın, given an atomless Borel probability measure $η$ supported on a compact subset of $\mathbb{R}$, Freiberg and Zähle introduced a measure-geometric approach to define a first order differential operator $\nabla_η$ and a second order differential operator $Δ_η$, with respect to $η$. We generalise this approach to measures of the form $η= ν+ δ$, where $ν$ is continuous and $δ$ is finitely supported. We determine analytic properties of $\nabla_η$ and $Δ_η$ and show that $Δ_η$ is a densely defined, unbounded, linear, self-adjoint operator with compact resolvent. Moreover, we give a systematic way to calculate the eigenvalues and eigenfunctions of $Δ_η$. For two leading examples, we determine the eigenvalues and the eigenfunctions, as well as the asymptotic growth rates of the eigenvalue counting function.

math.DS↗

arXiv2019

Intermediately trimmed strong laws for Birkhoff sums on subshifts of finite type

We prove strong laws of large numbers under intermediate trimming for Birkhoff sums over subshifts of finite type. This gives another application of a previous trimming result only proven for interval maps. In case of Markov measures we give a further example of St.\ Petersburg type distribution functions. To prove these statements we introduce the space of quasi-Hölder continuous functions for subshifts of finite type.

math.DS↗

arXiv2018

Limit theorems for counting large continued fraction digits

We establish a central limit theorem for counting large continued fraction digits $(a_n)$, i.e. we count occurrences $\{a_n>b_n\}$, where $(b_n)$ is a sequence of positive integers. Our result improves a similar result by Philipp which additionally assumes that $b_n$ tends to infinity. Moreover, we give a refinement of the famous Borel-Bernstein Theorem for continued fractions regarding the event that the $n$-th continued fraction digit lies infinitely often between $d_n$ and $d_n(1+1/c_n)$ for given sequences $(c_n)$ and $(d_n)$. Also for these sets we obtain a central limit theorem. As an interesting side result we determine the first $ϕ$-mixing coefficient for the Gauss system explicitly.

math.PR↗

arXiv2018

A multifractal analysis for cuspidal windings on hyperbolic surfaces

In this paper we investigate the multifractal decomposition of the limit set of a finitely generated, free Fuchsian group with respect to the mean cusp winding number. We will completely determine its multifractal spectrum by means of a certain free energy function and show that the Hausdorff dimension of sets consisting of limit points with the same scaling exponent coincides with the Legendre transform of this free energy function. As a by-product we generalise previously obtained results on the multifractal formalism for infinite iterated function systems to the setting of infinite graph directed Markov systems.

math.DS↗

arXiv2018

Measure-geometric Laplacians for discrete distributions

In 2002 Freiberg and Zähle introduced and developed a harmonic calculus for measure-geometric Laplacians associated to continuous distributions. We show their theory can be extended to encompass distributions with finite support and give a matrix representation for the resulting operators. In the case of a uniform discrete distribution we make use of this matrix representation to explicitly determine the eigenvalues and the eigenfunctions of the associated Laplacian.

math.DS↗

arXiv2018

Diffraction of return time measures

Letting $T$ denote an ergodic transformation of the unit interval and letting $f \colon [0,1)\to \mathbb{R}$ denote an observable, we construct the $f$-weighted return time measure $μ_y$ for a reference point $y\in[0,1)$ as the weighted Dirac comb with support in $\mathbb{Z}$ and weights $f \circ T^z(y)$ at $z\in\mathbb{Z}$, and if $T$ is non-invertible, then we set the weights equal to zero for all $z < 0$. Given such a Dirac comb, we are interested in its diffraction spectrum which emerges from the Fourier transform of its autocorrelation and analyse it for the dependence on the underlying transformation. For certain rapidly mixing transformations and observables of bounded variation, we show that the diffraction of $μ_{y}$ consists of a trivial atom and an absolutely continuous part, almost surely with respect to $y$. This contrasts what occurs in the setting of regular model sets arising from cut and project schemes and deterministic incommensurate structures. As a prominent example of non-mixing transformations, we consider the family of rigid rotations $T_α \colon x \to x + α\bmod{1}$ with rotation number $α\in \mathbb{R}^+$. In contrast to when $T$ is mixing, we observe that the diffraction of $μ_{y}$ is pure point, almost surely with respect to $y$. Moreover, if $α$ is irrational and the observable $f$ is Riemann integrable, then the diffraction of $μ_{y}$ is independent of $y$. Finally, for a converging sequence $(α_{i})_{i \in \mathbb{N}}$ of rotation numbers, we provide new results concerning the limiting behaviour of the associated diffractions.

math.DS↗

arXiv2018

The Sierpiński gasket as the Martin boundary of a non-isotropic Markov chain

In 2012 Lau and Ngai, motivated by the work of Denker and Sato, gave an example of an isotropic Markov chain on the set of finite words over a three letter alphabet, whose Martin boundary is homeomorphic to the Sierpiński gasket. Here, we extend the results of Lau and Ngai to a class of non-isotropic Markov chains. We determine the Martin boundary and show that the minimal Martin boundary is a proper subset of the Martin boundary. In addition, we give a description of the set of harmonic functions.

math.DS↗

arXiv2017

Strong laws of large numbers for intermediately trimmed sums of i.i.d. random variables with infinite mean

We consider moderately trimmed sums of non-negative i.i.d. random variables. We show that for every distribution function there exists a proper moderate trimming such that for the trimmed sum a non-trivial strong law of large numbers holds. In case that the distribution function has regularly varying tails we give necessary and sufficient conditions on the trimming for a strong law of large numbers to hold.

math.PR↗

arXiv2017

A classification of aperiodic order via spectral metrics and Jarník sets

Given an $α> 1$ and a $θ$ with unbounded continued fraction entries, we characterise new relations between Sturmian subshifts with slope $θ$ with respect to (i) an $α$-Höder regularity condition of a spectral metric, (ii) level sets defined in terms of the Diophantine properties of $θ$, and (iii) complexity notions which we call $α$-repetitive, $α$-repulsive and $α$-finite; generalisations of the properties known as linearly repetitive, repulsive and power free, respectively. We show that the level sets relate naturally to (exact) Jarn\'ık sets and prove that their Hausdorff dimension is $2/(α+ 1)$.

math.DS↗

arXiv2017

Regularity of aperiodic minimal subshifts

At the turn of this century Durand, and Lagarias and Pleasants established that key features of minimal subshifts (and their higher-dimensional analogues) to be studied are linearly repetitive, repulsive and power free. Since then, generalisations and extensions of these features, namely $α$-repetitive, $α$-repulsive and $α$-finite ($α\geq 1$), have been introduced and studied. We establish the equivalence of $α$-repulsive and $α$-finite for general subshifts over finite alphabets. Further, we studied a family of aperiodic minimal subshifts stemming from Grigorchuk's infinite $2$-group $G$. In particular, we show that these subshifts provide examples that demonstrate $α$-repulsive (and hence $α$-finite) is not equivalent to $α$-repetitive, for $α> 1$. We also give necessary and sufficient conditions for these subshifts to be $α$-repetitive, and $α$-repulsive (and hence $α$-finite). Moreover, we obtain an explicit formula for their complexity functions from which we deduce that they are uniquely ergodic.

math.DS↗

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

Escape rates for special flows and their higher order asymptotics

In this paper escape rates and local escape rates for special flows are sudied. In a general context the first result is that the escape rate depends monotonically on the ceiling function and fulfills certain scaling, invariance, and continuity properties. For the metric setting local escape rates are considered. If the base transformation is ergodic and exhibits an exponential convergence in probability of ergodic sums, then the local escape rate with respect to the flow is just the local escape rate with respect to the base transformation, divided by the integral of the ceiling function. Also a reformulation with respect to induced pressure is presented. Finally, under additional regularity conditions higher order asymptotics for the local escape rate are established.

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

Some recent developments in quantization of fractal measures

We give an overview on the quantization problem for fractal measures, including some related results and methods which have been developed in the last decades. Based on the work of Graf and Luschgy, we propose a three-step procedure to estimate the quantization errors. We survey some recent progress, which makes use of this procedure, including the quantization for self-affine measures, Markov-type measures on graph-directed fractals, and product measures on multiscale Moran sets. Several open problems are mentioned.

math.PR↗

arXiv2014

A note on measure-geometric Laplacians

We consider the measure-geometric Laplacians $Δ^μ$ with respect to atomless compactly supported Borel probability measures $μ$ as introduced by Freiberg and Zähle in 2002 and show that the harmonic calculus of $Δ^μ$ can be deduced from the classical (weak) Laplacian. We explicitly calculate the eigenvalues and eigenfunctions of $Δ^μ$. Further, it is shown that there exists a measure-geometric Laplacian whose eigenfunctions are the Chebyshev polynomials and illustrate our results through specific examples of fractal measures, namely Salem and inhomogeneous self-similar Cantor measures.

math.FA↗

arXiv2014

On the convergence to equilibrium of unbounded observables under a family of intermittent interval maps

We consider a family $\{ T_{r} \colon [0, 1] \circlearrowleft \}_{r \in [0, 1]}$ of Markov interval maps interpolating between the Tent map $T_{0}$ and the Farey map $T_{1}$. Letting $\mathcal{P}_{r}$ denote the Perron-Frobenius operator of $T_{r}$, we show, for $β\in [0, 1]$ and $α\in (0, 1)$, that the asymptotic behaviour of the iterates of $\mathcal{P}_{r}$ applied to observables with a singularity at $β$ of order $α$ is dependent on the structure of the $ω$-limit set of $β$ with respect to $T_{r}$. Having a singularity it seems that such observables do not fall into any of the function classes on which convergence to equilibrium has been previously shown.

math.DS↗