The pinwheel triangle of Conway and Radin is a standard example for tilings with self-similarity and statistical circular symmetry. Many modifications were constructed, all based on partitions of triangles or rectangles. The fractal example of Frank and Whittaker requires 13 different types of tiles. We present an example of a single tile with fractal boundary and very simple geometric structure which has the same symmetry and spectral properties as the pinwheel triangle.
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We study the attractors γ of finite graph directed systems S of contracting similarities in R d whose components are Jordan arcs. We prove that every self-similar Jordan arc different from a straight line segment may be partitioned into finitely many nonoverlapping subarcs δ j each of which also admits a partition into nonoverlapping images of subarcs δ j under contracting similarities. A formal description for this property is given by the multizipper construction.Although the bibliography on the geometry of self-similar sets is rather broad [1][2][3][4], the questions of structure description of these sets often fall beyond the range of vision of the authors. The subject of this article, which is a continuation of [5,6], concerns the description of the structure of self-similar Jordan arcs. An arc γ ∈ R n is called self-similar if there is a finite covering of γ by (generally, overlapping) closed subarcs γ 1 , . . . , γ m , each the union of the images of these subarcs under some contracting similarities. Some strict description of this property is given by means of the notion of a graph directed system of similarities which was introduced by Mauldin and Williams in [7]. The main result (Theorem 4.1) claims that we can replace such covering with a finite partition into nonoverlapping subarcs δ j each of which splits again into nonoverlapping images of the subarcs δ j under contracting similarities. A formal description of the last property is given by the multizipper construction of this article.In § 1 we collect some preliminaries, in § 2 we give the multizipper construction, § 3 is devoted to the properties of sequences of similarities of Jordan arcs, and in § 4 we formulate and prove the main theorem. The author expresses his gratitude to V. V. Aseev, A. S. Kravchenko, A. D. Mednykh, and A. V. Sychëv for useful remarks. § 1. Preliminaries 1.1. Directed graphs. Let G = (V, E) be a directed multigraph with vertices V and edges E. Given a pair u, v ∈ V , let E uv ⊆ E be the set of edges from u to v. If e ∈ E uv then u is called the initial vertex and v is the final vertex of e and the notations u = i * (e) and v = f * (e) are used.We require that each vertex u in the graph G be the initial vertex of at least one edge in G.A path σ in the graph G is a word σ = e 1 e 2 . . . e k , where e i ∈ E, such that f * (e i ) = i * (e i+1 ) for every i = 1, . . . , k − 1. Moreover, the vertex u = i * (e 1 ) is the initial vertex and v = f * (e k ) is the final vertex of σ. We use the same notations for the initial and final verteces of σ: u = i * (σ) and v = f * (σ). We denote by E (k) uv the set of paths σ from u to v of length k, i.e., all paths σ = e 1 e 2 . . . e k such thatuv is the set of all paths from u to v in G. Also, we put Ev . If σ = e 1 . . . e k is a path of length k and l < k then σ| l = e 1 . . . e l denotes the initial segment of length l in the word σ. We order each of these spaces by the relation on assuming that σ σ if σ is the prefix in σ . Similarly, σ is the predecessor of σ (σ σ ) if σ = e 1 . . . ...
We define G-symmetric polygonal systems of similarities and study the properties of symmetric dendrites, which appear as their attractors. This allows us to find the conditions under which the attractor of a zipper becomes a dendrite.
We study Möbius and quasimöbius mappings in spaces with a semimetric meeting the Ptolemy inequality. We construct a bimetrization of a Ptolemeic space which makes it possible to introduce a Möbius-invariant metric (angular distance) in the complement to each nonsingleton. This metric coincides with the hyperbolic metric in the canonical cases. We introduce the notion of generalized angle in a Ptolemeic space with vertices a pair of sets, determine its magnitude in terms of the angular distance and study distortion of generalized angles under quasimöbius embeddings. As an application to noninjective mappings, we consider the behavior of the generalized angle under projections and obtain an estimate for the inverse distortion of generalized angles under quasimeromorphic mappings (mappings with bounded distortion).
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