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Marc Kesseböhmer

Publications and source records attributed to Marc Kesseböhmer.

At least 55 records · Page 3Linked to original sources
arXiv2014

On the asymptotics of the $α$-Farey transfer operator

We study the asymptotics of iterates of the transfer operator for non-uniformly hyperbolic $α$-Farey maps. We provide a family of observables which are Riemann integrable, locally constant and of bounded variation, and for which the iterates of the transfer operator, when applied to one of these observables, is not asymptotic to a constant times the wandering rate on the first element of the partition $α$. Subsequently, sufficient conditions on observables are given under which this expected asymptotic holds. In particular, we obtain an extension theorem which establishes that, if the asymptotic behaviour of iterates of the transfer operator is known on the first element of the partition $α$, then the same asymptotic holds on any compact set bounded away from the indifferent fixed point.

math.DS↗
arXiv2014

The quantization for Markov-type measures on a class of ratio-specified graph directed fractals

We study the asymptotic quantization error of order $r$ for Markov-type measures $μ$ on a class of ratio-specified graph directed fractals. We show that the quantization dimension of $μ$ exists and determine its exact value $s_{r}$ in terms of spectral radius of a related matrix. We prove that the $s_{r}$-dimensional lower quantization coefficient of $μ$ is always positive. Moreover, inspired by Mauldin-Williams's work on the Hausdorff measure of graph directed fractals, we establish a necessary and sufficient condition for the $s_{r}$-dimensional upper quantization coefficient of $μ$ to be finite.

math.PR↗
arXiv2013

On the quantization for self-affine measures on Bedford-McMullen carpets

For a self-affine measure on a Bedford-McMullen carpet we prove that its quantization dimension exists and determine its exact value. Further, we give various sufficient conditions for the corresponding upper and lower quantization coefficient to be both positive and finite. Finally, we compare the quantization dimension with corresponding quantities derived from the multifractal temperature function and show that -- different from conformal systems -- they in general do not coincide.

math.MG↗
arXiv2012

A Fréchet law and an Erdös-Philipp law for maximal cuspidal windings

In this paper we establish a Fréchet law for maximal cuspidal windings of the geodesic flow on a Riemannian surface associated with an arbitrary finitely generated, essentially free Fuchsian group with parabolic elements. This result extends previous work by Galambos and Dolgopyat and is obtained by applying Extreme Value Theory. Subsequently, we show that this law gives rise to an Erdös-Philipp law and to various generalised Khintchine-type results for maximal cuspidal windings. These results strengthen previous results by Sullivan, Stratmann and Velani for Kleinian groups, and extend earlier work by Philipp on continued fractions, which was inspired by a conjecture of Erdös.

math.DS↗
arXiv2011

Spectral metric spaces for Gibbs measures

We construct spectral metric spaces for Gibbs measures on a one-sided topologically exact subshift of finite type. That is, for a given Gibbs measure we construct a spectral triple and show that Connes' corresponding pseudo-metric is a metric and that its metric topology agrees with the weak-*-topology on the state space over the set of continuous functions defined on the subshift. Moreover, we show that each Gibbs measure can be fully recovered from the noncommutative integration theory and that the noncommutative volume constant of the associated spectral triple is equal to the reciprocal of the measure theoretical entropy of the shift invariant Gibbs measure.

math.OA↗
arXiv2011

Multiresolution analysis for Markov Interval Maps

We set up a multiresolution analysis on fractal sets derived from limit sets of Markov Interval Maps. For this we consider the $\mathbb{Z}$-convolution of a non-atomic measure supported on the limit set of such systems and give a thorough investigation of the space of square integrable functions with respect to this measure. We define an abstract multiresolution analysis, prove the existence of mother wavelets, and then apply these abstract results to Markov Interval Maps. Even though, in our setting the corresponding scaling operators are in general not unitary we are able to give a complete description of the multiresolution analysis in terms of multiwavelets.

math.FA↗
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↗
arXiv2011

Strong renewal theorems and Lyapunov spectra for $α$-Farey and $α$-Lüroth systems

In this paper we introduce and study the $α$-Farey map and its associated jump transformation, the $α$-Lüroth map, for an arbitrary countable partition $α$ of the unit interval with atoms which accumulate only at the origin. These maps represent linearised generalisations of the Farey map and the Gauss map from elementary number theory. First, a thorough analysis of some of their topological and ergodic-theoretic properties is given, including establishing exactness for both types of these maps. The first main result then is to establish weak and strong renewal laws for what we have called $α$-sum-level sets for the $α$-Lüroth map. Similar results have previously been obtained for the Farey map and the Gauss map, by using infinite ergodic theory. In this respect, a side product of the paper is to allow for greater transparency of some of the core ideas of infinite ergodic theory. The second remaining result is to obtain a complete description of the Lyapunov spectra of the $α$-Farey map and the $α$-Lüroth map in terms of the thermodynamical formalism. We show how to derive these spectra, and then give various examples which demonstrate the diversity of their behaviours in dependence on the chosen partition $α$.

math.DS↗
arXiv2010

Induced topological pressure for countable state Markov shifts

We introduce the notion of induced topological pressure for countable state Markov shifts with respect to a non-negative scaling function and an arbitrary subset of finite words. Firstly, the scaling function allows a direct access to important thermodynamical quantities, which are usually given only implicitly by certain identities involving the classically defined pressure. In this context we generalise Savchenko's definition of entropy for special flows to a corresponding notion of topological pressure and show that this new notion coincides with the induced pressure for a large class of Hölder continuous height functions not necessarily bounded away from zero. Secondly, the dependence on the subset of words gives rise to interesting new results connecting the Gurevi{\vc} and the classical pressure with exhausting principles for a large class of Markov shifts. In this context we consider dynamical group extentions to demonstrate that our new approach provides a useful tool to characterise amenability of the underlying group structure.

math.DS↗
arXiv2010

A dichotomy between uniform distributions of the Stern-Brocot and the Farey sequence

We employ infinite ergodic theory to show that the even Stern-Brocot sequence and the Farey sequence are uniformly distributed mod 1 with respect to certain canonical weightings. As a corollary we derive the precise asymptotic for the Lebesgue measure of continued fraction sum-level sets as well as connections to asymptotic behaviours of geometrically and arithmetically restricted Poincaré series. Moreover, we give relations of our main results to elementary observations for the Stern-Brocot tree.

math.NT↗
arXiv2010

Regularity of multifractal spectra of conformal iterated function systems

We investigate multifractal regularity for infinite conformal iterated function systems (cIFS). That is we determine to what extent the multifractal spectrum depends continuously on the cIFS and its thermodynamic potential. For this we introduce the notion of regular convergence for families of cIFS not necessarily sharing the same index set, which guarantees the convergence of the multifractal spectra on the interior of their domain. In particular, we obtain an Exhausting Principle for infinite cIFS allowing us to carry over results for finite to infinite systems, and in this way to establish a multifractal analysis without the usual regularity conditions. Finally, we discuss the connections to the $λ$-topology introduced by Roy and Urbas{ń}ki.

math.DS↗
arXiv2009

Large deviations asymptotics for large waiting times

In this paper we investigate the statistics of large waiting times (with respect to the total waiting time) for Bernoulli processes. We determine the corresponding rate functions explicitly and prove a large deviations asymptotic. By this we have estabished a large deviation principle for which the rate function is not the Legendre transform of some free energy function.

math.PR↗
arXiv2009

Wavelets for iterated function systems

We construct a wavelet and a generalised Fourier basis with respect to some fractal measures given by one-dimensional iterated function systems. In this paper we will not assume that these systems are given by linear contractions generalising in this way some previous work of Jorgensen and Dutkay to the non-linear setting.

math.FA↗
arXiv2009

On the Lebesgue measure of sum-level sets for continued fractions

In this paper we give a detailed measure theoretical analysis of what we call sum-level sets for regular continued fraction expansions. The first main result is to settle a recent conjecture of Fiala and Kleban, which asserts that the Lebesgue measure of these level sets decays to zero, for the level tending to infinity. The second and third main result then give precise asymptotic estimates for this decay. The proofs of these results are based on recent progress in infinite ergodic theory, and in particular, they give non-trivial applications of this theory to number theory. The paper closes with a discussion of the thermodynamical significance of the obtained results, and with some applications of these to metrical Diophantine analysis.

math.DS↗
arXiv2009

The arithmetic-geometric scaling spectrum for continued fractions

To compare continued fraction digits with the denominators of the corresponding approximants we introduce the arithmetic-geometric scaling. We will completely determine its multifractal spectrum by means of a number theoretical free energy function and show that the Hausdorff dimension of sets consisting of irrationals with the same scaling exponent coincides with the Legendre transform of this free energy function. Furthermore, we identify the asymptotic of the local behaviour of the spectrum at the right boundary point and discuss a connection to the set of irrationals with continued fraction digits exceeding a given number which tends to infinity.

math.NT↗
arXiv2008

Sets of non-differentiability for conjugacies between expanding interval maps

We study differentiability of topological conjugacies between expanding piecewise $C^{1+ε}$ interval maps. If these conjugacies are not $C^1$, then they have zero derivative almost everywhere. We obtain the result that in this case the Hausdorff dimension of the set of points for which the derivative of the conjugacy does not exist lies strictly between zero and one. Using multifractal analysis and thermodynamic formalism, we show that this Hausdorff dimension is explicitly determined by the Lyapunov spectrum. Moreover, we show that these results give rise to a "rigidity dichotomy" for the type of conjugacies under consideration.

math.DS↗
arXiv2007

Hölder-differentiability of Gibbs distribution functions

In this paper we give non-trivial applications of the thermodynamic formalism to the theory of distribution functions of Gibbs measures (devil's staircases) supported on limit sets of finitely generated conformal iterated function systems in $\R$. For a large class of these Gibbs states we determine the Hausdorff dimension of the set of points at which the distribution function of these measures is not $α$-Hölder-differentiable. The obtained results give significant extensions of recent work by Darst, Dekking, Falconer, Li, Morris, and Xiao. In particular, our results clearly show that the results of these authors have their natural home within thermodynamic formalism.

math.DS↗