Connectedness of a class of two-dimensional self-affine tiles associated with triangular matrices

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• 作者：Jingcheng Liu^a ; ^{liujingcheng11@126.com" class="auth_mail" title="E-mail the corresponding author} ; Sze-Man Ngai^b ; ^c ; ^{smngai@georgiasouthern.edu" class="auth_mail" title="E-mail the corresponding author} ; Juan Tao^b ; ^{tj_zk001@163.com" class="auth_mail" title="E-mail the corresponding author}
• 关键词：Self-affine tile ; Digit set ; Connectedness
• 刊名：Journal of Mathematical Analysis and Applications
• 出版年：2016
• 出版时间：15 March 2016
• 年：2016
• 卷：435
• 期：2
• 页码：1499-1513
• 全文大小：1098 K

文摘

We study the connectedness of planar self-affine sets T(A,D)$T(A,\mathcal{D})$ generated by a matrix of the form a059677a7ec1d980aafc6ead2b2c36">$A=\left[\begin{array}{cc}\hfill p\hfill & \hfill 0\hfill \\ \hfill -a\hfill & \hfill q\hfill \end{array}\right]$ together with nonconsecutive and noncollinear digit sets of the form D={l₀,l₁,…,l_|p|−1}×{m₀,m₁,…,m_|q|−1}$\mathcal{D}=\{l_0,l_1,\dots ,l_{|p|-1}\}\times \{m_0,m_1,\dots ,m_{|q|-1}\}$, where {l₀,l₁,…,l_|p|−1}$\{l_0,l_1,\dots ,l_{|p|-1}\}$ and 08a22a6333602779de1a617224e3d3a" title="Click to view the MathML source">{m₀,m₁,…,m_|q|−1}$\{m_0,m_1,\dots ,m_{|q|-1}\}$ are residue systems for |p|$|p|$ and |q|$|q|$ respectively. We give a necessary and sufficient condition for T(A,D)$T(A,\mathcal{D})$ to be connected, and extend some results by Deng and Lau (2011) [5] to nonconsecutive digit sets.