108 research outputs found

    On the connectedness of planar self-affine sets

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    In this paper, we consider the connectedness of planar self-affine set T(A,D)T(A,\mathcal{D}) arising from an integral expanding matrix AA with characteristic polynomial f(x)=x2+bx+cf(x)=x^2+bx+c and a digit set D={0,1,,m}v\mathcal{D}=\{0,1,\dots, m\}v. The necessary and sufficient conditions only depending on b,c,mb,c,m are given for the T(A,D)T(A,\mathcal{D}) to be connected. Moreover, we also consider the case that D{\mathcal D} is non-consecutively collinear.Comment: 18 pages; 18 figure

    Connectedness of planar self-affine sets associated with non-consecutive collinear digit sets

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    In the paper, we focus on the connectedness of planar self-affine sets T(A,D)T(A,{\mathcal{D}}) generated by an integer expanding matrix AA with det(A)=3|\det (A)|=3 and a collinear digit set D={0,1,b}v{\mathcal{D}}=\{0,1,b\}v, where b>1b>1 and vR2v\in {\mathbb{R}}^2 such that {v,Av}\{v, Av\} is linearly independent. We discuss the domain of the digit bb to determine the connectedness of T(A,D)T(A,{\mathcal{D}}). Especially, a complete characterization is obtained when we restrict bb to be an integer. Some results on the general case of det(A)>3|\det (A)|> 3 are obtained as well.Comment: 15 pages, 10 figure

    Boundaries of Disk-like Self-affine Tiles

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    Let T:=T(A,D)T:= T(A, {\mathcal D}) be a disk-like self-affine tile generated by an integral expanding matrix AA and a consecutive collinear digit set D{\mathcal D}, and let f(x)=x2+px+qf(x)=x^{2}+px+q be the characteristic polynomial of AA. In the paper, we identify the boundary T\partial T with a sofic system by constructing a neighbor graph and derive equivalent conditions for the pair (A,D)(A,{\mathcal D}) to be a number system. Moreover, by using the graph-directed construction and a device of pseudo-norm ω\omega, we find the generalized Hausdorff dimension dimHω(T)=2logρ(M)/logq\dim_H^{\omega} (\partial T)=2\log \rho(M)/\log |q| where ρ(M)\rho(M) is the spectral radius of certain contact matrix MM. Especially, when AA is a similarity, we obtain the standard Hausdorff dimension dimH(T)=2logρ/logq\dim_H (\partial T)=2\log \rho/\log |q| where ρ\rho is the largest positive zero of the cubic polynomial x3(p1)x2(qp)xqx^{3}-(|p|-1)x^{2}-(|q|-|p|)x-|q|, which is simpler than the known result.Comment: 26 pages, 11 figure

    Rational self-affine tiles

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    An integral self-affine tile is the solution of a set equation AT=dD(T+d)\mathbf{A} \mathcal{T} = \bigcup_{d \in \mathcal{D}} (\mathcal{T} + d), where A\mathbf{A} is an n×nn \times n integer matrix and D\mathcal{D} is a finite subset of Zn\mathbb{Z}^n. In the recent decades, these objects and the induced tilings have been studied systematically. We extend this theory to matrices AQn×n\mathbf{A} \in \mathbb{Q}^{n \times n}. We define rational self-affine tiles as compact subsets of the open subring Rn×pKp\mathbb{R}^n\times \prod_\mathfrak{p} K_\mathfrak{p} of the ad\'ele ring AK\mathbb{A}_K, where the factors of the (finite) product are certain p\mathfrak{p}-adic completions of a number field KK that is defined in terms of the characteristic polynomial of A\mathbf{A}. Employing methods from classical algebraic number theory, Fourier analysis in number fields, and results on zero sets of transfer operators, we establish a general tiling theorem for these tiles. We also associate a second kind of tiles with a rational matrix. These tiles are defined as the intersection of a (translation of a) rational self-affine tile with Rn×p{0}Rn\mathbb{R}^n \times \prod_\mathfrak{p} \{0\} \simeq \mathbb{R}^n. Although these intersection tiles have a complicated structure and are no longer self-affine, we are able to prove a tiling theorem for these tiles as well. For particular choices of digit sets, intersection tiles are instances of tiles defined in terms of shift radix systems and canonical number systems. Therefore, we gain new results for tilings associated with numeration systems
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