‎We prove that the historic set for ratio‎ ‎of Birkhoff average is either empty or full of Hausdorff dimension in a class of one dimensional‎ ‎non-uniformly hyperbolic dynamical systems.

Making an extensive use of small transfinite topological dimension trind, we ascribe to every metric space X an ordinal number (or −1 or Ω) tHD(X), and we call it the transfinite Hausdorff dimension of X. This ordinal number shares many common features with Hausdorff dimension. It is monotone with respect to subspaces, it is invariant under bi-Lipschitz maps (but in general not under homeomorph...

It is possible for one to define fractals as those sets which have non-integral Hausdorff Dimension. This paper defines Hausdorff Dimension and rigorously introduces the necessary theory to prove that one such set is indeed a fractal. The first chapter includes a proof of Banach’s Fixed Point Theorem. The Hausdorff Metric is defined and it is mentioned how one produces ‘generators’ for creating...

We continue the study of effective Hausdorff dimension as it was initiated by LUTZ. Whereas he uses a generalization of martingales on the Cantor space to introduce this notion we give a characterization in terms of effective s-dimensional Hausdorff measures, similar to the effectivization of Lebesgue measure by MARTIN-LÖF. It turns out that effective Hausdorff dimension allows to classify sequ...

In this paper we establish a characterization of abelian compact Hausdorff semigroups with unique idempotent and ideal retraction property. We also introduce a function algebra on a semitopological semigroup whose associated semigroup compactification is universal withrespect to these properties.

Our motivation for the study of negligible subsets is that, for a compact piecewise linear n-manifold M, it has been a long standing problem as to whether the space of homeomorphisms H(M) is an absolute neighborhood retract (The Homeomorphism Group Problem). It turns out ([GH]) that there is a dense Gδ subset G ⊂ H(M) which is homeomorphic to a s-manifold, where s is the countable infinite prod...

In this paper, we study the concepts of Wijsman statistical convergence, Hausdorff statistical convergence and  Wijsman statistical Cauchy double sequences of sets and investigate the relationship between them.

We have collected definitions and basic results for the (centered ball) density in metric space with respect to an arbitrary Hausdorff function. We have kept the definitions general: we do not assume the Hausdorff functions are continuous or blanketed, and we do not assume the metric space is a subset of Euclidean space. We discuss the covering measure (= centered Hausdorff measure) and packing...

In an earlier work, joint with R. Kenyon, we computed the Hausdorff dimension of the “multiplicative golden mean shift” defined as the set of all reals in [0, 1] whose binary expansion (xk) satisfies xkx2k = 0 for all k ≥ 1. Here we show that this set has infinite Hausdorff measure in its dimension. A more precise result in terms of gauges in which the Hausdorff measure is infinite is also obta...