Luke G. Rogers

Derivations and Dirichlet forms on fractals

Abstract We study derivations and Fredholm modules on metric spaces with a local regular conservative Dirichlet form. In particular, on finitely ramified fractals, we show that there is a non-trivial Fredholm module if and only if the fractal is not a tree (i.e. not simply connected). This result relates Fredholm modules and topology, refines and improves known results on p.c.f. fractals. We also discuss weakly summable Fredholm modules and the Dixmier trace in the cases of some finitely and infinitely ramified fractals (including non-self-similar fractals) if the so-called spectral dimension is less than 2. In the finitely ramified self-similar case we relate the p -summability question with estimates of the Lyapunov exponents for harmonic functions and the behavior of the pressure function.

Laplacians on the basilica Julia set

We consider the basilica Julia set of the polynomial

Estimates for the resolvent kernel of the Laplacian on p.c.f. self-similar fractals and blowups

P(z)=z^{2}-1

Gradients of Laplacian eigenfunctions on the Sierpinski gasket

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

Embedding convex geometries and a bound on convex dimension

2

Smooth bumps, a Borel theorem and partitions of smooth functions on p.c.f. fractals

, the exponent in the Weyl law is

Pseudo-differential operators on fractals and other metric measure spaces.

\log9/\log6

Generalized Eigenfunctions and a Borel Theorem on the Sierpinski Gasket

, and we can compute all the eigenvalues and eigenfunctions by a spectral decimation method. The other is graph-directed self-similar under the map

Power dissipation in fractal AC circuits

z\mapsto P(z)

THE RESOLVENT KERNEL FOR PCF SELF-SIMILAR FRACTALS

; it has energy renormalization factor