Harmonic analysis on the Pascal graph (Q1019706)

The spectrum of \(\Delta\) in \(\ell^2(\Gamma)\) is determined for the Pascal graph \(\Gamma\), the infinite, connected planar 3-regular graph, and for some compactification of \(\Gamma\). The eigenvalues are related to the Julia set of some one-dimensional real map \(f\) and the preimages \(f^{-n}(0)\). The eigenfunctions as well as the orthogonal complement of the eigenspaces in \(\ell^2\) are described very precisely. Moreover, the results can be transferred to the infinite, connected planar 4-regular Sierpińsky graph.