The Spectrum of the Laplacian on the Pentagasket

2003 ◽  
pp. 1-24 ◽  
Author(s):  
Bryant Adams ◽  
S. Alex Smith ◽  
Robert S. Strichartz ◽  
Alexander Teplyaev
Keyword(s):  
Spectrum Of The Laplacian

On the spectrum of the Laplacian with frequently alternating boundary conditions

1999 ◽  
Vol 118 (3) ◽  
pp. 272-277 ◽  
Author(s):  
D. I. Borisov ◽  
R. R. Gadyl’shin
Keyword(s):  
Boundary Conditions ◽  
Spectrum Of The Laplacian

scholarly journals SPECTRUM OF THE LAPLACIAN OF AN ASYMMETRIC FRACTAL GRAPH

2006 ◽  
Vol 49 (1) ◽  
pp. 101-113 ◽  
Author(s):  
Jonathan Jordan
Keyword(s):  
Similar Sequence ◽  
Graph Laplacians ◽  
Self Similar ◽  
Spectrum Of The Laplacian

AbstractWe consider a simple self-similar sequence of graphs that does not satisfy the symmetry conditions that imply the existence of a spectral decimation property for the eigenvalues of the graph Laplacians. We show that, for this particular sequence, a very similar property to spectral decimation exists, and we obtain a complete description of the spectra of the graphs in the sequence.

scholarly journals SPECTRUM OF THE LAPLACIAN ON THE VICSEK SET “WITH NO LOOSE ENDS”

Fractals ◽  
2017 ◽  
Vol 25 (06) ◽  
pp. 1750062 ◽  
Author(s):  
ROBERT S. STRICHARTZ ◽  
JIANGYUE ZHU
Keyword(s):  
Spectral Properties ◽  
Lipschitz Functions ◽  
Countable Number ◽  
Line Segments ◽  
Spectrum Of The Laplacian

We study the spectral properties of a fractal the Vicsek set “with no loose ends” (VNLE) obtained from the standard Vicsek set (VS) by making a countable number of identifications of points so that all the line segments in VS become circles in VNLE. We show that the standard Laplacian on VNLE satisfies spectral decimation with the same cubic renormalization polynomial as for VS, and thereby give a complete description of all eigenfunctions of the Laplacian. We then study the restrictions of eigenfunctions to the large circles in VNLE and prove that these are Lipschitz functions.

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