Hui Rao

Iterated Function Systems with Overlaps and Self-Similar Measures

The paper considers the iterated function systems of similitudes which satisfy a separation condition weaker than the open set condition, in that it allows overlaps in the iteration. Such systems include the well-known Bernoulli convolutions associated with the PV numbers, and the contractive similitudes associated with integral matrices. The latter appears frequently in wavelet analysis and the theory of tilings. One of the basic questions is studied: the absolute continuity and singularity of the self-similar measures generated by such systems. Various conditions to determine the dichotomy are given.

Atomic surfaces, tilings and coincidence I. irreducible case

An irreducible Pisot substitution defines a graph-directed iterated function system. The invariant sets of this iterated function system are called the atomic surfaces. In this paper, a new tiling of atomic surfaces, which contains Thurston’sβ-tiling as a subclass, is constructed. Related tiling and dynamical properties are studied. Based on the coincidence condition defined by Dekking [Dek], we introduce thesuper-coincidence condition. It is shown that the super-coincidence condition governs the tiling and dynamical properties of atomic surfaces. We conjecture that every Pisot substitution satisfies the super-coincidence condition.

On the structures and dimensions of Moran sets

AbstractThe Moran sets and the Moran class are defined by geometric fashion that distinguishes the classical self-similar sets from the following points:(i)The placements of the basic sets at each step of the constructions can be arbitrary.(ii)The contraction ratios may be different at each step.(iii)The lower limit of the contraction ratios permits zero. The properties of the Moran sets and Moran class are studied, and the Hausdorff, packing and upper Box-counting dimensions of the Moran sets are determined by net measure techniques. It is shown that some important properties of the self-similar sets no longer hold for Moran sets.

On the open set condition for self-similar fractals

For self-similar sets, the existence of a feasible open set is a natural separation condition which expresses geometric as well as measure-theoretic properties. We give a constructive approach by defining a central open set and characterizing those points which do not belong to feasible open sets.

TOPOLOGICAL STRUCTURE OF SELF-SIMILAR SETS

We consider the attractor T of injective contractions f1, …, fm on R2 which satisfy the Open Set Condition. If T is connected, then Ts interior T° is either empty or has no holes, and Ts boundary ∂T is connected; if further T° is non-empty and connected, then ∂T is a simple closed curve, thus T is homeomorphic to the unit disk {x∈R2: |x|≤1}.

Topology and separation of self-similar fractals in the plane

Even though the open set condition (OSC) is generally accepted as the right condition to control overlaps of self-similar sets, it seems unclear how it relates to the actual size of the overlap. For connected self-similar sets in the plane, we prove that a finite overlap implies OSC. On the other hand, there are Cantor sets with arbitrary small dimensions which do not fulfil the OSC.

Purely periodic beta-expansions with Pisot unit base

Let β > 1 be a Pisot unit. A family of sets {X i } 1≤i≤q defined by a β-numeration system has been extensively studied as an atomic surface or Rauzy fractal. For the purpose of constructing a Markov partition, a domain X = U q i =1 X i constructed by an atomic surface has appeared in several papers. In this paper we show that the domain X completely characterizes the set of purely periodic β-expansions.

Expanding Polynomials and Connectedness of Self-Affine Tiles

Abstract Little is known about the connectedness of self-affine tiles in

Gap sequence, Lipschitz equivalence and box dimension of fractal sets

{\Bbb R}^n

On substitution Invariant Sturmian Words: An Application of Rauzy Fractals

. In this note we consider this property on the self-affine tiles that are generated by consecutive collinear digit sets. By using an algebraic criterion, we call it the {\it height reducing property}, on expanding polynomials (i.e., all the roots have moduli