arXiv subjects

Marc Kesseböhmer

Publications and source records attributed to Marc Kesseböhmer.

63 records · Page 4Linked to original sources

arXiv2007

Fractal analysis for sets of non-differentiability of Minkowski's question mark function

In this paper we study various fractal geometric aspects of the Minkowski question mark function $Q.$ We show that the unit interval can be written as the union of the three sets $Λ_{0}:=\{x:Q'(x)=0\}$, $Λ_{\infty}:=\{x:Q'(x)=\infty\}$, and $Λ_{\sim}:=\{x:Q'(x)$ does not exist and $Q'(x)\not=\infty\}.$ The main result is that the Hausdorff dimensions of these sets are related in the following way. $\dim_{H}(ν_{F})<\dim_{H}(Λ_{\sim})= \dim_{H} (Λ_{\infty}) = \dim_{H} (\mathcal{L}(h_{\mathrm{top}}))<\dim_{H}(Λ_{0})=1.$ Here, $\mathcal{L}(h_{\mathrm{top}})$ refers to the level set of the Stern-Brocot multifractal decomposition at the topological entropy $h_{\mathrm{top}}=\log2$ of the Farey map $F,$ and $\dim_{H}(ν_{F})$ denotes the Hausdorff dimension of the measure of maximal entropy of the dynamical system associated with $F.$ The proofs rely partially on the multifractal formalism for Stern-Brocot intervals and give non-trivial applications of this formalism.

math.DS↗

arXiv2007

Higher-dimensional multifractal value sets for conformal infinite graph directed Markov systems

We give a description of the level sets in the higher dimensional multifractal formalism for infinite conformal graph directed Markov systems. If these systems possess a certain degree of regularity this description is complete in the sense that we identify all values with non-empty level sets and determine their Hausdorff dimension. This result is also partially new for the finite alphabet case.

math.DS↗

arXiv2007

Large deviation asymptotics for continued fraction expansions

We study large deviation asymptotics for processes defined in terms of continued fraction digits. We use the continued fraction digit sum process to define a stopping time and derive a joint large deviation asymptotic for the upper and lower fluctuation process. Also a large deviation asymptotic for single digits is given.

math.NT↗

arXiv2006

A distributional limit law for the continued fraction digit sum

We consider the continued fraction digits as random variables measured with respect to Lebesgue measure. The logarithmically scaled and normalized fluctuation process of the digit sums converges strongly distributional to a random variable uniformly distributed on the unit interval. For this process normalized linearly we determine a large deviation asymptotic.

math.NT↗

arXiv2006

Limiting modular symbols and their fractal geometry

In this paper we use fractal geometry to investigate boundary aspects of the first homology group for finite coverings of the modular surface. We obtain a complete description of algebraically invisible parts of this homology group. More precisely, we first show that for any modular subgroup the geodesic forward dynamic on the associated surface admits a canonical symbolic representation by a finitely irreducible shift space. We then use this representation to derive an `almost complete' multifractal description of the higher--dimensional level sets arising from Manin--Marcolli's limiting modular symbols.

math.GT↗

arXiv2006

Radon--Nikodym representations of Cuntz--Krieger algebras and Lyapunov spectra for KMS states

We study relations between $(H,β)$--KMS states on Cuntz--Krieger algebras and the dual of the Perron--Frobenius operator $\mathcal{L}_{-βH}^{*}$. Generalising the well--studied purely hyperbolic situation, we obtain under mild conditions that for an expansive dynamical system there is a one--one correspondence between $(H,β)$--KMS states and eigenmeasures of $\mathcal{L}_{-βH}^{*}$ for the eigenvalue 1. We then consider representations of Cuntz--Krieger algebras which are induced by Markov fibred systems, and show that if the associated incidence matrix is irreducible then these are $\ast$--isomorphic to the given Cuntz--Krieger algebra. Finally, we apply these general results to study multifractal decompositions of limit sets of essentially free Kleinian groups $G$ which may have parabolic elements. We show that for the Cuntz--Krieger algebra arising from $G$ there exists an analytic family of KMS states induced by the Lyapunov spectrum of the analogue of the Bowen--Series map associated with $G$. Furthermore, we obtain a formula for the Hausdorff dimensions of the restrictions of these KMS states to the set of continuous functions on the limit set of $G$. If $G$ has no parabolic elements, then this formula can be interpreted as the singularity spectrum of the measure of maximal entropy associated with $G$.

math.DS↗

arXiv2006

Refined measurable rigidity and flexibility for conformal iterated function systems

In this paper we investigate aspects of rigidity and flexibility for conformal iterated function systems. For the case in which the systems are not essentially affine we show that two such systems are conformal equivalent if and only if in each of their Lyapunov spectra there exists at least one level set such that the corresponding Gibbs measures coincide. We then proceed by comparing this result with the essentially affine situation. We show that essentially affine systems are far less rigid than non--essentially affine systems, and subsequently we then investigate the extent of their flexibility.

math.DS↗

arXiv2006

Limit laws for distorted return time processes for infinite measure preserving transformations

We consider conservative ergodic measure preserving transformations on infinite measure spaces and investigate the asymptotic behaviour of distorted return time processes with respect to sets satisfying a type of Darling-Kac condition. We identify two critical cases for which we prove uniform distribution laws. For this we introduce the notion of uniformly returning sets and discuss some of their properties.

math.DS↗

arXiv2005

A multifractal analysis for Stern-Brocot intervals, continued fractions and Diophantine growth rates

In this paper we obtain multifractal generalizations of classical results by Lévy and Khintchin in metrical Diophantine approximations and measure theory of continued fractions. We give a complete multifractal analysis for Stern--Brocot intervals, for continued fractions and for certain Diophantine growth rates. In particular, we give detailed discussions of two multifractal spectra closely related to the Farey map and the Gauss map.

math.NT↗