Shorted operators: An application in potential theory

On nested fractals a “Laplacian” can be constructed as a scaled limit of difference operators. The appropriate scaling and starting configuration are given by a nonlinear, finite dimensional eigenvalue problem. We study it as a fixed point problem using Hilbertʹs projective metric on cones, a nonlinear generalization of the Perron-Frobenius theory of nonnegative matrices. The nonlinearity arises from a map Φ known as the shorted operator. Potential theoretic notions and results apply to it, since it acts on a cone of discrete “Laplacians” or difference operators. Usually, Φ is considered on the larger cone of positive semidefinite operators. We are able to take advantage of the more specific structure of the reduced domain because several properties of Φ are local. Results are possible with respect to continuity, concavity, the Fréchet derivative, invariant subcones, the geometry of these cones, and the contraction of Hilbertʹs metric.