We introduce a graph-directed pair of planar self-similar sets that possess fully symmetric Laplacians. For these two fractals, due to Shima’s celebrated criterion, we point out that one admits the spectral decimatio...We introduce a graph-directed pair of planar self-similar sets that possess fully symmetric Laplacians. For these two fractals, due to Shima’s celebrated criterion, we point out that one admits the spectral decimation by the canonic graph approximation and the other does not. For the second fractal, we adjust to choosing a new graph approximation guided by the directed graph, which still admits spectral decimation. Then we make a full description of the Dirichlet and Neumann eigenvalues and eigenfunctions of both of these two fractals.展开更多
In this paper, using spectral decimation, we prove that the "hot spots" conjecture holds on a class of homogeneous hierarchical gaskets introduced by Hambly,i.e., every eigenfunction of the second-smallest e...In this paper, using spectral decimation, we prove that the "hot spots" conjecture holds on a class of homogeneous hierarchical gaskets introduced by Hambly,i.e., every eigenfunction of the second-smallest eigenvalue of the Neumann Laplacian(introduced by Kigami) attains its maximum and minimum on the boundary.展开更多
基金supported by National Natural Science Foundation of China(Grant No.12071213)the Natural Science Foundation of Jiangsu Province in China(Grant No.BK20211142)。
文摘We introduce a graph-directed pair of planar self-similar sets that possess fully symmetric Laplacians. For these two fractals, due to Shima’s celebrated criterion, we point out that one admits the spectral decimation by the canonic graph approximation and the other does not. For the second fractal, we adjust to choosing a new graph approximation guided by the directed graph, which still admits spectral decimation. Then we make a full description of the Dirichlet and Neumann eigenvalues and eigenfunctions of both of these two fractals.
基金supported in part by NSFC grants Nos.11271327, 11771391
文摘In this paper, using spectral decimation, we prove that the "hot spots" conjecture holds on a class of homogeneous hierarchical gaskets introduced by Hambly,i.e., every eigenfunction of the second-smallest eigenvalue of the Neumann Laplacian(introduced by Kigami) attains its maximum and minimum on the boundary.
