The paper investigates fixed points and attractors of infinite iterated function systems in Cantor space. By means of the theory of formal languages simple examples of the non-coincidence of fixed point and attractor (closure of the fixed point) are given.

We propose the metric notion of strong hyperbolicity as a way of obtaining hyperbolicity with sharp additional properties. Specifically, strongly hyperbolic spaces are Gromov hyperbolic spaces that are metrically well-behaved at infinity, and, under weak geodesic assumptions, they are strongly bolic as well. We show that CAT(−1) spaces are strongly hyperbolic. On the way, we determine the best ...

In [Berarducci-Otero, 2004], the authors define an analogue of Lebesgue measure for bounded definable subsets of an o-minimal structure K , which expands a real closed field. Our aim is defining the area of a bounded definable d-dimensional subsets of Kn. Different possible definitions for such area are possible: we will prove that some of them are equivalent. On the reals, there are many diffe...

Following a recent paper [10] we show that the finiteness of square function associated with the Riesz transforms with respect to Hausdorff measure H (n is interger) on a set E implies that E is rectifiable.

We study how the Hausdorff measure is distributed in nonsymmetric narrow cones in Rn. As an application, we find an upper bound close to n−k for the Hausdorff dimension of sets with large k-porosity. With k-porous sets we mean sets which have holes in k different directions on every small scale.

We investigate the energy of arrangements of N points on a rectifiable d-dimensional manifold A ⊂ Rd′ that interact through the power law (Riesz) potential V = 1/r, where s > 0 and r is Euclidean distance in R ′ . With Es(A, N) denoting the minimal energy for such N -point configurations, we determine the asymptotic behavior (as N → ∞) of Es(A, N) for each fixed s ≥ d. Moreover, if A has positi...

We consider, for G a simply connected domain and 0 < p < ∞, the Hardy space H(G) formed by fixing a Riemann map τ of the unit disc onto G, and demanding of functions F holomorphic on G that the integrals of |F |p over the curves τ({|z| = r}) be bounded for 0 < r < 1. The resulting space is usually not the one obtained from the classical Hardy space of the unit disc by conformal mapping. This is...

We show that Hausdorff measures of different dimensions are not Borel isomorphic; that is, the measure spaces (R, B, H) and (R, B, H) are not isomorphic if s 6= t, s, t ∈ [0, 1], where B is the σ-algebra of Borel subsets of R and H is the d-dimensional Hausdorff measure. This answers a question of B. Weiss and D. Preiss. To prove our result, we apply a random construction and show that for ever...

For K ⊂ C compact, we say that K has vanishing analytic capacity (or γ(K) = 0) when all bounded analytic functions on C\K are constant. We would like to characterize γ(K) = 0 geometrically. Easily, γ(K) > 0 when K has Hausdorff dimension larger than 1, and γ(K) = 0 when dim(K) < 1. Thus only the case when dim(K) = 1 is interesting. So far there is no characterization of γ(K) = 0 in general, but...

There are two fundamental results in the classical theory of metric Diophantine approximation: Khintchine’s theorem and Jarńık’s theorem. The former relates the size of the set of well approximable numbers, expressed in terms of Lebesgue measure, to the behavior of a certain volume sum. The latter is a Hausdorff measure version of the former. We start by discussing these theorems and show that ...