H\"older-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 $\alpha$-H\"older-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.Comment: 13 pages, 2 figure

On the asymptotics of the α\alpha-Farey transfer operator

We study the asymptotics of iterates of the transfer operator for non-uniformly hyperbolic $\alpha$-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 $\alpha$. 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 $\alpha$, then the same asymptotic holds on any compact set bounded away from the indifferent fixed point

Constructing Restricted Patterson Measures for Geometrically Infinite Kleinian Groups

In this paper, we study exhaustions, referred to as ρ-restrictions, of arbitrary nonelementary Kleinian groups with at most finitely many bounded parabolic elements. Special emphasis is put on the geometrically infinite case, where we obtain that the limit set of each of these Kleinian groups contains an infinite family of closed subsets, referred to as ρ-restricted limit sets, such that there is a Poincaré series and hence an exponent of convergence δρ, canonically associated with every element in this family. Generalizing concepts which are well known in the geometrically finite case, we then introduce the notion of ρ-restricted Patterson measure, and show that these measures are non-atomic, δρ-harmonic, δρ-subconformal on special sets and δρ-conformal on very special sets. Furthermore, we obtain the results that each ρ-restriction of our Kleinian group is of δρ-divergence type and that the Hausdorff dimension of the ρ-restricted limit set is equal to δ

Convergence analysis of generalized iteratively reweighted least squares algorithms on convex function spaces

The computation of robust regression estimates often relies on minimization of a convex functional on a convex set. In this paper we discuss a general technique for a large class of convex functionals to compute the minimizers iteratively which is closely related to majorization-minimization algorithms. Our approach is based on a quadratic approximation of the functional to be minimized and includes the iteratively reweighted least squares algorithm as a special case. We prove convergence on convex function spaces for general coercive and convex functionals F and derive geometric convergence in certain unconstrained settings. The algorithm is applied to TV penalized quantile regression and is compared with a step size corrected Newton-Raphson algorithm. It is found that typically in the first steps the iteratively reweighted least squares algorithm performs significantly better, whereas the Newton type method outpaces the former only after many iterations. Finally, in the setting of bivariate regression with unimodality constraints we illustrate how this algorithm allows to utilize highly efficient algorithms for special quadratic programs in more complex settings. --regression analysis,monotone regression,quantile regression,shape constraints,L1 regression,nonparametric regression,total variation semi-norm,reweighted least squares,Fermat's problem,convex approximation,quadratic approximation,pool adjacent violators algorithm