Narrow Results

We consider the attractor T of injective contractions f₁, …, f_m on R² which satisfy the Open Set Condition. If T is connected, then T's interior T° is either empty or has no holes, and T's 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}.

We study sets of the form T ∩ g(T), where T is self-similar and g is a bijection. Under certain assumptions, this intersection may be described using a digraph IFS.

Recently, a lot of work has been devoted to the study of the Lipschitz equivalence between self-similar sets. Many related results on self-similar sets of the real line have been well-known under the condition that the similarity ratios equal each other. In this paper, we generalize some results to a more general setting. We mainly study the Lipschitz equivalence between two general Sierpinski carpets in ℝ².

It is well-known that the Hausdorff dimension of the Sierpinski triangle Λ is s = log 3/log 2. However, it is a long standing open problem to compute the s-dimensional Hausdorff measure of Λ denoted by . In the literature the best existing estimate is

The average distance between points of a fractal is proposed as a natural measure of the way in which the points of a fractal are distributed. The average distance between points of the Cantor Set is found to be , the average distance between points of the Cantor p-set is , and the average distance between points of the Fat Cantor p-set is . A general formula for computing the average distance between points of a self-similar set satisfying the open set condition is found.

Computing the Hausdorff measure of C × C, where C is the classical ternary Cantor set, is a long standing difficult problem. It is well-known that for a self-similar set, calculating the Hausdorff measure is equivalent to determining its optimal sets.

This paper studies optimal sets of C × C: their diameters, measures, symmetries and the shapes. For this purpose, we introduce several devices: the repulsive principle, a bipartite graph G and a W-function. We show that the diameter of the optimal set B is between 1.2993 and 1.3082. Two symmetry properties of B are proved. Finally, we show that the shape of B is very close to a disk. We conjecture that an optimal set might be a disk.

We establish a formula yielding the Hausdorff measure for a class of non-self-similar Cantor sets in terms of the canonical covers of the Cantor set.

This paper studies the gap sequences of graph-directed sets satisfying the strong separation condition. An interesting application is to investigate the gap sequences of self-similar sets with overlaps.

Self-similarity and Lipschitz equivalence are two basic and important properties of fractal sets. In this paper, we consider those properties of the union of Cantor set and its translate. We give a necessary and sufficient condition that the union is a self-similar set. Moreover, we show that the union satisfies the strong separation condition if it is of the self-similarity. By using the augment tree, we characterize the Lipschitz equivalence between Cantor set and the union of Cantor set and its translate.

For an iterated function system (IFS) with contractive similarities, the post-critically finite (p.c.f.) condition and H-condition are two important separation conditions in the literature. But their relationship is not clear. In this paper, we show that if the similarity matrices of the IFS are commensurable, then the p.c.f. condition implies H-condition.

Skeleton is a new notion designed for constructing space-filling curves of self-similar sets. In a previous paper by Dai and the authors [Space-filling curves of self-similar sets (II): Edge-to-trail substitution rule, Nonlinearity32(5) (2019) 1772–1809] it was shown that for all the connected self-similar sets with a skeleton satisfying the open set condition, space-filling curves can be constructed. In this paper, we give a criterion of existence of skeletons by using the so-called neighbor graph of a self-similar set. In particular, we show that a connected self-similar set satisfying the finite-type condition always possesses skeletons: an algorithm is obtained here.

For self-similar set $E={\lambda }_{1}E\cup ({\lambda }_{2}E+1-{\lambda }_2)$$({\lambda }_1+{\lambda }_2<1),$ we study the arithmetic product of $E$ and obtain that when ${\lambda }_1\ge {\lambda }_2>0$ then

Let $K$ be a self-similar set in $ℝ$. Generally, if the iterated function system (IFS) of $K$ has some overlaps, then some points in $K$ may have multiple codings. If an $x\in K$ has a unique coding, then we call $x$ a univoque point. We denote by ${𝒰}$ (univoque set) the set of points in $K$ having unique codings. In this paper, we shall consider the following natural question: if two self-similar sets are bi-Lipschitz equivalent, then are their associated univoque sets also bi-Lipschitz equivalent. We give a class of self-similar sets with overlaps, and answer the above question affirmatively.

The stretched Cantor product is not self-similar but rectifiable. Using the method of finite pattern, we study the mean geodesic distance on the stretched Cantor product.

The study of Lipschitz equivalence of fractals is a very active topic in recent years. Most of the studies in literature concern totally disconnected fractals. In this paper, using neighbor automata of self-similar sets and a transducer, we construct a bi-Lipschitz map between two fractal squares which are not totally disconnected. This is the first nontrivial construction of this type. We also show that this map is measure-preserving.

Wen [Some problems in fractal geometry, in Report in Chinese Conference on Fractal Geometry and Dynamical Systems, 2023] posed the following questions: when a quadrangle is a self-similar set with the open set condition (OSC). It is well known that any convex quadrangle is a self-similar set. For the concave quadrangles, this case is more complicated. Loosely speaking, some concave polygons may be not self-similar. In this paper, we prove two results. First, we give a necessary and sufficient condition such that a concave quadrangle is a self-similar set. Moreover, we also give a necessary condition for a concave quadrangle to be a self-similar set with the OSC. For the convex quadrangle, we investigate the trapezoid. We show under some condition that the trapezoid is a self-similar set with the OSC. In particular, for any trapezoid, if the ratio of the lengths of two bases is rational, then the trapezoid is a self-similar set with the OSC.

The authors show that the self-similar set for a finite family of contractive similitudes (similarities, i.e., |f_i(x) - f_i(y)| = a_i|x - y|, x,y ∈ R^N, where 0 < a_i < 1) is uniformly perfect except the case that it is singleton. As a corollary, it is proved that this self-similar set has positive Hausdorff dimension provided that it is not a singleton. And a lower bound of the upper box dimension of the uniformly perfect sets is given. Meanwhile the uniformly perfect set with Hausdorff measure zero in its Hausdorff dimension is given.