Basilica Conformal Measure Spectra
We have numerical results for the standard Basilica (c=-1) on graph levels 2-13, and variant (c=-1+.15i) on level 2-10. This page contains eigenvalue tables and pictures of the eigenfunctions.
The highlights of our findings:
Unlike for equilibrium measure, the spectrum of our Lapcian is not a topological invariant, thus we present results for both the standard basilica and another Julia Set with its topology (c=-1+.15i). This spectrum lacks much of the structure of the equilibrium measure. Though we continue to see function with horizontal or vertical support, we do not see derived functions. Also for a given eigenfunction, its location in the spectrum varies as we increase the level making predicting eigenvalues much more challenging.
- Here is a table of eigenvalues:
- Using the first 256 calculated eigenvalues on the highest level we get pictures of the Eigenvalue counting function and Weyl Ratio on each.
- Here are pictures of eigenfunctions on level 6-11 (6-10 for the variant). The HTML version contains the first 52 at each level, the PDF contains the first 256 eigenfunctions, where available. Each row of pictures corresponds to a single eigenfunction, the first column is the ray angle description of the function, the second is the support of function colored by the sign of function, the third shows the restriction of the function to the inner circle.
- Standard Basilica
- Variant Basilica: c=-1+.15c
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Last Update: 28 May 2009