arXiv 0802.32482008-02math.CAmath.DS

拉普拉斯在大教堂朱利亚集

Laplacians on the basilica Julia set

Luke G. Rogers, Alexander Teplyaev

备注：: 24 pages, one figure in separate eps file. Replaced the theorem that was removed in the second version, with a corrected proof. Corrected an error in the description of the energy forms (we are grateful to Jun Kigami for pointing out this error). Other minor changes
发表信息：: Commun. Pure Appl. Anal. 9 (2010), no. 1, 211-231
DOI：: 10.3934/cpaa.2010.9.211

摘要：

我们考虑了多项式$P(z)=z^{2}-1$的巴西利卡朱利亚集，并构造了所有可能的阻力（Dirichlet）形式，以及相应的拉普拉斯算子，使有效阻力度量中的拓扑与通常的拓扑一致.然后，我们集中讨论两个特殊情况。一种是自相似谐振结构，其能量重整化因子为2，谱维数为$\log9/\log6$，我们可以通过谱抽取方法计算所有的本征值和本征函数。另一种是图定向自相似映射$z\mapsto P(z)$，它具有能量重整化因子$\sqrt2$和4/3的谱维数，但谱的精确计算比较困难.后一种狄利克雷形式和拉普拉斯算子在某种意义上在巴西利卡朱利亚集上是共形不变的。

英文摘要：

We consider the basilica Julia set of the polynomial $P(z)=z^{2}-1$ and construct all possible resistance (Dirichlet) forms, and the corresponding Laplacians, for which the topology in the effective resistance metric coincides with the usual topology. Then we concentrate on two particular cases. One is a self-similar harmonic structure, for which the energy renormalization factor is 2, the spectral dimension is $\log9/\log6$, and we can compute all the eigenvalues and eigenfunctions by a spectral decimation method. The other is graph-directed self-similar under the map $z\mapsto P(z)$; it has energy renormalization factor $\sqrt2$ and spectral dimension 4/3, but the exact computation of the spectrum is difficult. The latter Dirichlet form and Laplacian are in a sense conformally invariant on the basilica Julia set.

机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。