Spectral Properties of Self-similar Lattices and Iteration of Rational MapsIn this text, the author considers discrete Laplace operators defined on lattices based on finitely ramified self-similar sets and their continuous analogs defined on the self-similar sets. He focuses on the spectral properties of these operators. The basic example is the lattice based on the Sierpinski gasket. He introduces a new renormalization map that appears to be a rational map defined on a smooth projective variety. (More precisely, this variety is isomorphic to a product of three types of Grassmannians: complex Grassmannians, Lagrangian Grassmannian, and orthogonal Grassmannians.) He relates some characteristics of the dynamics of its iterates with some characteristics of the spectrum of the operator. Specifically, he gives an explicit formula for the density of states in terms of the Green current of the map, and he relates the indeterminacy points of the map with the so-called Neumann-Dirichlet eigenvalues which lead to eigenfunctions with compact support on the unbounded lattice. Depending on the asymptotic degree of the map, he can prove drastically different spectral properties of the operators. The formalism is valid for the general class of finitely ramified self-similar sets. |
Contents
JIANFor DUKUNS | |
nstruction of a selfsimilar Laplacian 17 | |
me basic results 23 | |
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Common terms and phrases
algebraically stable analytic subset appendix basis bidegree blow-up canonical Ck+1 codimension cohomology compact complex manifolds consider coordinates defined definition degree denote density Dirichlet form discrete Laplace operator eigenfunctions eigenvalues embedding equal exists Fatou set finite formula G-invariant Grassmann algebra Green current Hence holomorphic function hypersurface ImHf implies indeterminacy points integer irreducible isomorphic iterates Julia set Kähler form Kähler manifold Lagrangian Grassmannian lattice Lelong number lemma map g matrix measure Neumann-Dirichlet null OF(n order of vanishing orthogonal projective space proof proposition prove psh function pull-back Q in S+ rational map ReHf relation remark renormalization renormalization map resp scalar product self-similar self-similar sets sequence Sierpinski gasket spectrum subspace subvariety SymG symmetric symmetric matrix Symp theorem 4.3 u₁ v₁ µND