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    "description": "Abstract.Let T = T(A, D) be a self-affine attractor in \n\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym}\n\t\t\t\t\\usepackage{amsfonts}\n\t\t\t\t\\usepackage{amssymb}\n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}\n\t$$ \\mathbb{R}^n $$\n\t\\end{document} \ndefined by an integral expanding matrix A and\na digit set D. In the\nfirst part of this paper, in connection with canonical number systems,\nwe study connectedness of T when \nD corresponds to the set of\nconsecutive integers \n\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym}\n\t\t\t\t\\usepackage{amsfonts}\n\t\t\t\t\\usepackage{amssymb}\n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}\n\t$$ \\{0, 1,\\ldots, |\\det(A)| - 1\\}  $$\n\t\\end{document} \n. It is shown that in \n\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym}\n\t\t\t\t\\usepackage{amsfonts}\n\t\t\t\t\\usepackage{amssymb}\n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}\n\t$$ \\mathbb{R}^3 $$\n\t\\end{document}  \nand \n\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym}\n\t\t\t\t\\usepackage{amsfonts}\n\t\t\t\t\\usepackage{amssymb}\n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}\n\t$$ \\mathbb{R}^4 $$\n\t\\end{document}\n, for any integral expanding matrix A, T(A, D) is connected.\n\nIn the second part, we study connectedness of Pisot dual tiles, which\nplay an important role in the study of \n\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym}\n\t\t\t\t\\usepackage{amsfonts}\n\t\t\t\t\\usepackage{amssymb}\n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}\n\t$$ \\beta $$\n\t\\end{document}\n-expansions, substitutions and\nsymbolic dynamical systems. It is shown that each tile of the dual\ntiling generated by a Pisot unit of degree 3 is arcwise connected. This\nis naturally expected since the digit set consists of consecutive\nintegers as above. However surprisingly, we found families of\ndisconnected Pisot dual tiles of degree 4. We even give a simple\nnecessary and sufficient condition of connectedness of the Pisot dual\ntiles of degree 4. Detailed proofs will be given in [4].", 
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    "name": "On the connectedness of self-affine attractors", 
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