Marc Kesseböhmer

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

Abstract 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 to the Gauss map.

A multifractal formalism for growth rates and applications to geometrically finite Kleinian groups

We elaborate thermodynamic and multifractal formalisms for general classes of potential functions and their average growth rates. We then apply these formalisms to certain geometrically finite Kleinian groups which may have parabolic elements of different ranks. We show that for these groups our revised formalisms give access to a description of the spectrum of ‘homological growth rates’ in terms of Hausdorff dimension. Furthermore, we derive necessary and sufficient conditions for the existence of ‘thermodynamic phase transitions’.

Strong renewal theorems and Lyapunov spectra for alpha-Farey and alpha-Luroth systems

In this paper, we introduce and study the -Farey map and its associated jump transformation, the -Luroth map, for an arbitrary countable partition of the unit interval with atoms which accumulate only at the origin. These maps represent linearized generalizations of the Farey map and the Gauss map from elementary number theory. First, a thorough analysis of some of their topological and ergodic theoretical 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 -sum-level sets for the -Luroth 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 -Farey map and the -Luroth 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 .

Fractal curvature measures and Minkowski content for self-conformal subsets of the real line

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 C diffeomorphic image. Both in the lattice and nonlattice situation average versions of the fractal curvature measures are shown to always exist. 1. Brief Introduction Notions of curvature are an important tool to describe the geometric structure of sets and have been introduced and intensively studied for broad classes of sets. Originally, the idea to characterise sets in terms of their curvature stems from the study of smooth manifolds as well as from the theory of convex bodies with sufficiently smooth boundaries. In his fundamental paper Curvature Measures [Fed59], Federer localises, extends and unifies the previously existing notions of curvature to sets of positive reach. This is where he introduces curvature measures, which can be viewed as a measure theoretical substitute for the notion of curvature for sets without a differentiable structure. Federer’s curvature measures were studied and generalised in various ways. An extension to finite unions of convex bodies is given in [Gro78] and [Sch80] and to finite unions of sets with positive reach in [Zäh84]. In [Win08], Winter extends the curvature measures to fractal sets in R, which typically cannot be expressed as finite unions of sets with positive reach. These measures are referred to as fractal curvature measures and are defined as weak limits of rescaled versions of the curvature measures introduced by Federer, Groemer and Schneider. Winter also examines conditions for their existence in the self-similar case. However, fractal sets arising in geometry (for instance as limit sets of Fuchsian groups) or in number theory (for instance as sets defined by Diophantine inequalities) are typically non self-similar but rather self-conformal. In order Date: March 9, 2011. 2000 Mathematics Subject Classification. Primary 28A80; Secondary 28A75, 60K05.

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

Abstract 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 ∞ : = { x : Q ′ ( x ) = ∞ } , and Λ ∼ : = { x : Q ′ ( x ) does not exist and Q ′ ( x ) ≠ ∞ } . The main result is that the Hausdorff dimensions of these sets are related in the following way: dim H ( ν F ) dim H ( Λ ∼ ) = dim H ( Λ ∞ ) = dim H ( L ( h top ) ) dim H ( Λ 0 ) = 1 . Here, L ( h top ) refers to the level set of the Stern–Brocot multifractal decomposition at the topological entropy h top = log 2 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.

Induced topological pressure for countable state Markov shifts

We generalise Savchenkos definition of topological entropy for special flows over countable Markov shifts by considering the corresponding notion of topological pressure. For a large class of Holder continuous height functions not necessarily bounded away from zero, this pressure can be expressed by our new 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, and we are able to set up a variational principle in this context. Investigating the dependence of induced pressure on the subset of words, we give interesting new results connecting the Gurevic and the classical pressure with exhaustion principles for a large class of Markov shifts. In this context we consider dynamical group extensions to demonstrate that our new approach provides a useful tool to characterise amenability of the underlying group structure.

Wavelets for iterated function systems

Abstract We construct a wavelet and a generalised Fourier basis with respect to some fractal measure given by a one-dimensional iterated function system. In this paper we will not assume that these systems are given by linear contractions generalising in this way some previous work of Dutkay, Jorgensen, and Pedersen to the non-linear setting. As a byproduct we are able to provide a Fourier basis also for such linear fractals like the Middle Third Cantor Set which have been left out by previous approaches.

STERN–BROCOT PRESSURE AND MULTIFRACTAL SPECTRA IN ERGODIC THEORY OF NUMBERS

In this note we apply the general multifractal analysis for growth rates derived in [10], and show that this leads to some new results in ergodic theory and the theory of multifractals of numbers. Namely, we consider Stern–Brocot growth rates and introduce the Stern–Brocot pressure P. We then obtain the results that P is differentiable everywhere and that its Legendre transformation governs the multifractal spectra arising from level sets of Stern–Brocot rates.

Thermodynamic formalism, large deviation, and multifractals

The spectral gap of the transfer operator for an expanding dynamical systems relates large deviation and local limit theorems. We discuss this phenomenon and state a local large deviation theorem in symbolic dynamical systems due to the second author ([8]). This general viewpoint also implies the multifractal formalism for topological Markov chains.

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 no- tion 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 Urba´ nski.