A Survey on the Topology of Fractal Squares
We consider a special type of self-similar sets, called fractal squares, and give a brief review on recent results and unsolved issues with an emphasis on their topological properties.
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arXiv subjects
Hui Rao
Publications and source records attributed to Hui Rao.
Point-wise doubling indices of measures and its application to bi-Lipschitz classification of Bedford-McMullen carpets
Doubling measure was introduced by Beurling and Ahlfors in 1956 and now it becomes a basic concept in analysis on metric space. In this paper, for a measure which is not doubling, we introduce a notion of point-wise doubling index, and calculate the point-wise doubling indices of uniform Bernoulli measures on Bedford-McMullen carpets. As an application, we show that, except a small class of Bedford-McMullen carpets, if two Bedford-McMullen carpets are bi-Lipschitz equivalent, then they have the same fiber sequence up to a permutation.
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Lattice subsequences of fixed points of Toeplitz substitutions
We define the modulo-$m$ Toeplitz fixed point generated by Toeplitz substitution and study the lattice subsequence of such fixed point. Moreover, we provide a method to check whether one modulo-$m$ Toeplitz fixed point is a lattice subsequence of another.
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Characterization of self-affine tile digit sets on $\mathbb{R}^n$
Let $R$ be an $n\times n$ expanding matrix with integral entries. A fundamental question in the fractal tiling theory is to understand the structure of the digit set $\mathcal{D}\subset\mathbb{Z}^n$ so that the integral self-affine set $T(R,\mathcal{D})$ is a translational tile on $\mathbb{R}^n$ In this paper, we introduce a notion of skew-product-form digit set which is a very general class of tile digit sets. Especially, we show that in the one-dimensional case, if $T(b,\mathcal{D})$ is a self-similar tile, then there exists $m\geq 1$ such that $$\mathcal{D}_m=\mathcal{D}+b\mathcal{D}+\cdots+b^{m-1}\mathcal{D}$$ is a skew-product-form digit set. Notice that $T(b,\mathcal{D})=T(b^m,\mathcal{D}_m)$, in some sense, we completely characterize the self-similar tiles in $\mathbb{R}^1$ As an application, we establish that all self-similar tiles $T(b,\mathcal{D})$ where $b=p^{\alpha}q^{\beta}$ contains at most two prime factors are spectral sets in $\mathbb{R}^1$.
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Topology automaton and conformal dimension of post-critical-finite self-similar sets
In this paper, we use a class of finite state automata, called topology automaton, to study the metric classification of a special class of post-critically finite self-similar sets. As an application, we prove that the conformal dimension of post-critically finite self-similar dendrites and fractal gasket with connected component is 1.
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Box-counting measure of metric spaces
In this paper, we introduce a new notion called the \emph{box-counting measure} of a metric space. We show that for a doubling metric space, an Ahlfors regular measure is always a box-counting measure; consequently, if $E$ is a self-similar set satisfying the open set condition, then the Hausdorff measure restricted to $E$ is a box-counting measure. We show two classes of self-affine sets, the generalized Lalley-Gatzouras type self-affine sponges and Barański carpets, always admit box-counting measures; this also provides a very simple method to calculate the box-dimension of these fractals. Moreover, among others, we show that if two doubling metric spaces admit box-counting measures, then the multi-fractal spectra of the box-counting measures coincide provided the two spaces are Lipschitz equivalent.
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Invariance of multifractal spectrum of uniform self-affine measures and its applications
We study the bi-Lipschitz classification of Bedford-McMullen carpets which are totally disconnected. Let $E$ be a such carpet and let $μ_E$ be the uniform Bernoulli measure on $E$. We show that the multifractal spectrum and the doubling property of $μ_E$ are both invariant under a bi-Lipschitz map. Moreover, we show that if $μ_E$ and $μ_F$ are doubling, then a bi-Lipschitz map between $E$ and $F$ enjoys a certain measure preserving property.
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Ergodicity for $p$-adic continued fraction algorithms
Following Schweiger's generalization of multidimensional continued fraction algorithms, we consider a very large family of $p$-adic multidimensional continued fraction algorithms, which include Schneider's algorithm, Ruban's algorithms, and the $p$-adic Jacobi-Perron algorithm as special cases. The main result is to show that all the transformations in the family are ergodic with respect to the Haar measure.
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A dimension drop phenomenon of fractal cubes
Let E be a metric space. We introduce a notion of connectedness index of E, which is the Hausdor? dimension of the union of non-trivial connected components of E. We show that the connectedness index of a fractal cube E is strictly less than the Hausdor? dimension of E provided that E possesses a trivial connected component. Hence the connectedness index is a new Lipschitz invariant. Moreover, we investigate the relation between the connectedness index and topological Hausdor? dimension.
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Characterization of complementing pairs of $({\mathbb Z}_{\geq 0})^n$
Let $A, B, C$ be subsets of an abelian group $G$. A pair $(A, B)$ is called a $C$-pair if $A, B\subset C$ and $C$ is the direct sum of $A$ and $B$. The $(\Z_{\geq 0})$-pairs are characterized by de Bruijn in 1950 and the $(\Z_{\geq 0})^2$-pairs are characterized by Niven in 1971. In this paper, we characterize the $(\Z_{\geq 0})^n$-pairs for all $n\geq 1$. We show that every $(\Z_{\geq 0})^n$-pair is characterized by a weighted tree if it is primitive, that is, it is not a Cartesian product of a $(\Z_{\geq 0})^p$-pair and a $(\Z_{\geq 0})^q$-pair of lower dimensions.
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Dissecting a square into congruent polygons
We study the dissection of a square into congruent convex polygons. Yuan \emph{et al.} [Dissecting the square into five congruent parts, Discrete Math. \textbf{339} (2016) 288-298] asked whether, if the number of tiles is a prime number $\geq 3$, it is true that the tile must be a rectangle. We conjecture that the same conclusion still holds even if the number of tiles is an odd number $\geq 3$. Our conjecture has been confirmed for triangles in earlier works. We prove that the conjecture holds if either the tile is a convex $q$-gon with $q\geq 6$ or it is a right-angle trapezoid.
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Space-filling curves of self-similar sets (III): Skeletons
Skeleton is a new notion designed for constructing space-filling curves of self-similar sets. It is shown in [Dai, Rao and Zhang, Space-filling curves of self-similar sets (II): Edge-to-trail substitution rule,https://doi.org/10.1088/1361-6544/ab1275] that for a connected self-similar set, space-filling curves can be constructed provided that it possesses a skeleton. 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.
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Space-filling curves of self-similar sets (II): Edge-to-trail substitution rule
It is well-known that the constructions of space-filling curves depend on certain substitution rules. For a given self-similar set, finding such rules is somehow mysterious, and it is the main concern of the present paper. Our first idea is to introduce the notion of skeleton for a self-similar set. Then, from a skeleton, we construct several graphs, define edge-to-trail substitution rules, and explore conditions ensuring the rules lead to space-filling curves. Thirdly, we summarize the classical constructions of the space-filling curves into two classes: the traveling-trail class and the positive Euler-tour class. Finally, we propose a general Euler-tour method, using which we show that if a self-similar set satisfies the open set condition and possesses a skeleton, then space-filling curves can be constructed. Especially, all connected self-similar sets of finite type fall into this class. Our study actually provides an algorithm to construct space-filling curves of self-similar sets.
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Every component of a fractal square is a Peano continuum
This paper concerns the local connectedness of components of self-similar sets. Given an equal partition of the unit square into n*n small squares, we may choose arbitrarily two or more of them and form an iterated function system. The attractor F resulted from this IFS is called a fractal square. We prove that every component of F is locally connected. The same result for three-dimensional analogues of F does not hold.
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Lipschitz equivalence of fractals and finite state automaton
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 finite state automata, we construct a bi-Lipschitz map between two fractal squares which are not totally disconnected. This is the first non-trivial map of this type. We also show that this map is measure-preserving.
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Lipschitz invariance of walk dimension on connected self-similar sets
Walk dimension is an important conception in analysis of fractals. In this paper we prove that the walk dimension of a connected compact set possessing an Alfors regular measure is an invariant under Lipschitz transforms. As an application, we show some generalized Sierpiński gaskets are not Lipschitz equivalent.
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Space-filling curves of self-similar sets (I): Iterated function systems with order structure
This paper is the first paper of three papers in a series, which intend to provide a systematic treatment for the space-filling curves of self-similar sets. In the present paper, we introduce a notion of \emph{linear graph-directed IFS} (linear GIFS in short). We show that to construct a space-filling curve of a self-similar set, it is amount to explore its linear GIFS structures. Some other notions, such as chain condition, path-on-lattice IFS, and visualizations of space-filling curves are also concerned. In sequential papers \cite{Dai15} and \cite{RZ14}, we obtain a universal algorithm to construct space-filling curves of self-similar sets of finite type, that is, as soon as the IFS is given, the computer will do everything automatically. Our study extends almost all the known results on space-filling curves.
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Higher dimensional Frobenius problem and Lipschitz equivalence of Cantor sets
The higher dimensional Frobenius problem was introduced by a preceding paper [Fan, Rao and Zhang, Higher dimensional Frobenius problem: maximal saturated cones, growth function and rigidity, Preprint 2014]. %the higher dimensional Frobenius problem was introduced and a directional growth function was studied. In this paper, we investigate the Lipschitz equivalence of dust-like self-similar sets in $\mathbb R^d$. For any self-similar set, we associate with it a higher dimensional Frobenius problem, and we show that the directional growth function of the associate higher dimensional Frobenius problem is a Lipschitz invariant. As an application, we solve the Lipschitz equivalence problem when two dust-like self-similar sets $E$ and $F$ have coplanar ratios, by showing that they are Lipschitz equivalent if and only if the contraction vector of the $p$-th iteration of $E$ is a permutation of that of the $q$-th iteration of $F$ for some $p, q\geq 1$. This partially answers a question raised by Falconer and Marsh [On the Lipschitz equivalence of Cantor sets, \emph{Mathematika,} \textbf{39} (1992), 223--233].
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