Laplacians on the basilica Julia set

Abstract

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 $\log 9/\log 6$, 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 $\sqrt{2}$ 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.

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

Cited by in corpus (14)

• Derivations and Dirichlet forms on fractals
• Dirac and magnetic Schrödinger operators on fractals
• Local Dirichlet forms, Hodge theory, and the Navier-Stokes equations on topologically one-dimensional fractals
• Spectral dimension and Bohr's formula for Schrodinger operators on unbounded fractal spaces
• A discrete Gauss-Green identity for unbounded Laplace operators and transience of random walks
• Spectral properties of graphs associated to the Basilica group
• Spectrum of the Laplacian on the Vicsek Set "with no loose ends"
• Existence and uniqueness of diffusions on the Julia sets of Misiurewicz-Sierpinski maps
• Abelian Sandpile Model on Randomly Rooted Graphs and Self-Similar Groups
• Dirichlet forms on unconstrained Sierpinski carpets
• Brownian motion on the golden ratio Sierpinski gasket
• "Graph Paper" Trace Characterizations of Functions of Finite Energy
• Dual graphs and modified Barlow--Bass resistance estimates for repeated barycentric subdivisions
• Laplacians on Julia sets of rational maps