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Hausdorff dimension in graph directed constructions
We introduce the notion of geometric constructions in Rm governed by a directed graph G and by similarity ratios which are labelled with the edges of this graph. For each such construction, we
Dimensions and Measures in Infinite Iterated Function Systems
The Hausdorff and packing measures and dimensions of the limit sets of iterated function systems generated by countable families of conformal contractions are investigated. Conformal measures for
Graph Directed Markov Systems: Geometry and Dynamics of Limit Sets
Introduction 1. Symbolic dynamics 3. Holder families of functions 4. Conformal graph directed Markov systems 5. Examples of graph directed Markov systems 6. Conformal iterated function systems 7.
Random recursive constructions: asymptotic geometric and topological properties
We study some notions of "random recursive constructions" in Euclidean m-space which lead almost surely to a particular type of topological object; e.g., Cantor set, Sierpiriski curve or Menger
Multifractal decompositions of Moran fractals
We present a rigorous construction and generalization of the multifractal decomposition for Moran fractals with infinite product measure. The generalization is specified by a system of nonnegative
Gibbs states on the symbolic space over an infinite alphabet
We consider subshifts of finite type on the symbolic space generated by incidence matrices over a countably infinite alphabet. We extend the definition of topological pressure to this context and, as
On the Hausdorff dimension of some graphs
Consider the functions Wb(x)= b-cn[1(bnX + On)--1(0n)] n=-oo where b > 1, 0 0 such that if b is large enough, then the Hausdorff dimension of the graph of Wb is bounded below by 2a (C/ ln b). We also
Conformal iterated function systems with applications to the geometry of continued fractions
In this paper we obtain some results about general conformal iterated function systems. We obtain a simple characterization of the packing dimension of the limit set of such systems and introduce
Zero Temperature Limits of Gibbs-Equilibrium States for Countable Alphabet Subshifts of Finite Type
Let ΣA be a finitely primitive subshift of finite type on a countable alphabet. For appropriate functions f:ΣA → IR, the family of Gibbs-equilibrium states (μtf)t⩾1 for the functions tf is shown to
Some additive properties of sets of real numbers
Some problems concerning the additive properties of subsets of R are investigated. From a result of G. G . Lorentz in additive number theory, we show that if P is a nonempty perfect subset of R, then