Algorithmic Randomness and Capacity of Closed Sets

We investigate the connection between measure, capacity and algorithmic randomness for the space of closed sets. For any computable measure m, a computable capacity T may be defined by letting T(Q) be the measure of the family of closed sets K which have nonempty intersection with Q. We prove an effective version of Choquet's capacity theorem by showing that every computable capacity may be obtained from a computable measure in this way. We establish conditions on the measure m that characterize when the capacity of an m-random closed set equals zero. This includes new results in classical probability theory as well as results for algorithmic randomness. For certain computable measures, we construct effectively closed sets with positive capacity and with Lebesgue measure zero. We show that for computable measures, a real q is upper semi-computable if and only if there is an effectively closed set with capacity q

Level Sets of the Takagi Function: Local Level Sets

The Takagi function \tau : [0, 1] \to [0, 1] is a continuous non-differentiable function constructed by Takagi in 1903. The level sets L(y) = {x : \tau(x) = y} of the Takagi function \tau(x) are studied by introducing a notion of local level set into which level sets are partitioned. Local level sets are simple to analyze, reducing questions to understanding the relation of level sets to local level sets, which is more complicated. It is known that for a "generic" full Lebesgue measure set of ordinates y, the level sets are finite sets. Here it is shown for a "generic" full Lebesgue measure set of abscissas x, the level set L(\tau(x)) is uncountable. An interesting singular monotone function is constructed, associated to local level sets, and is used to show the expected number of local level sets at a random level y is exactly 3/2.Comment: 32 pages, 2 figures, 1 table. Latest version has updated equation numbering. The final publication will soon be available at springerlink.co

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.Comment: 13 page

Wavelets techniques for pointwise anti-Holderian irregularity

In this paper, we introduce a notion of weak pointwise Holder regularity, starting from the de nition of the pointwise anti-Holder irregularity. Using this concept, a weak spectrum of singularities can be de ned as for the usual pointwise Holder regularity. We build a class of wavelet series satisfying the multifractal formalism and thus show the optimality of the upper bound. We also show that the weak spectrum of singularities is disconnected from the casual one (denoted here strong spectrum of singularities) by exhibiting a multifractal function made of Davenport series whose weak spectrum di ers from the strong one

Variable Length Markov Chains, Persistent Random Walks: a close encounter

Book chapterThis is the story of the encounter between two worlds: the world of random walks and the world of Variable Length Markov Chains (VLMC). The meeting point turns around the semi-Markov property of underlying processes