There is a well established multifractal theory for self-similar measures generated by non-overlapping contractive similutudes. Our report here concerns those with overlaps. In particular we restrict our at...There is a well established multifractal theory for self-similar measures generated by non-overlapping contractive similutudes. Our report here concerns those with overlaps. In particular we restrict our attention to the important classes of self-similar measures that have matrix representations. The dimension spectra and the L q -spectra are analyzed through the product of matrices. There are abnormal behaviors on the multifractal structure and they will be discussed in detail.展开更多
We define a new Markov chain on the symbolic space representing the Sierpinski gasket (SG),and show that the corresponding Martin boundary is homeomorphic to the SG while the minimal Martin boundary is the three verti...We define a new Markov chain on the symbolic space representing the Sierpinski gasket (SG),and show that the corresponding Martin boundary is homeomorphic to the SG while the minimal Martin boundary is the three vertices of the SG.In addition,the harmonic structure induced by the Markov chain coincides with the canonical one on the SG.This suggests another approach to consider the existence of Laplacians on those self-similar sets for which the problem is still not settled.展开更多
文摘There is a well established multifractal theory for self-similar measures generated by non-overlapping contractive similutudes. Our report here concerns those with overlaps. In particular we restrict our attention to the important classes of self-similar measures that have matrix representations. The dimension spectra and the L q -spectra are analyzed through the product of matrices. There are abnormal behaviors on the multifractal structure and they will be discussed in detail.
文摘We define a new Markov chain on the symbolic space representing the Sierpinski gasket (SG),and show that the corresponding Martin boundary is homeomorphic to the SG while the minimal Martin boundary is the three vertices of the SG.In addition,the harmonic structure induced by the Markov chain coincides with the canonical one on the SG.This suggests another approach to consider the existence of Laplacians on those self-similar sets for which the problem is still not settled.
