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Spectral analysis on Barlow and Evans’ projective limit fractals
Journal of Spectral Theory ( IF 1 ) Pub Date : 2021-02-24 , DOI: 10.4171/jst/337
Benjamin Steinhurst 1 , Alexander Teplyaev 2
Affiliation  

We review the projective limit construction of a state space for a Markov process use by Barlow and Evans. On this state space we construct a projective limit Dirichlet form in a process analogous to Barlow and Evan’s construction of a Markov process. Then we study the spectral properties of the corresponding Laplacian using the projective limit construction. For some examples, such as the Laakso spaces and a Sierpiński pâte à choux, one can develop a complete spectral theory, including the eigenfunction expansions that are analogous to Fourier series. In addition, we construct connected fractal spaces isospectral to the fractal strings of Lapidus and van Frankenhuijsen. Our work is motivated by recent progress in mathematical physics on fractals.

中文翻译:

Barlow和Evans的投影极限分形的光谱分析

我们回顾了Barlow和Evans使用的马尔可夫过程的状态空间的投影极限构造。在此状态空间上,我们以类似于Barlow和Evan的马尔可夫过程的构造过程构造投影极限Dirichlet形式。然后,我们使用射影极限构造研究相应拉普拉斯算子的光谱特性。对于某些示例,例如Laakso空间和Sierpińskipâteàchoux,可以开发出完整的光谱理论,包括类似于傅立叶级数的本征函数展开式。此外,我们构造了与Lapidus和van Frankenhuijsen的分形串等谱的连通分形空间。我们的工作受到分形数学物理学的最新进展的启发。
更新日期:2021-03-17
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