Simplicial Prefractals and Topological Calculus

Topological behaviour of self-similar spectra for fractal domains is shown. Two different mathematical tools are employed: the theory of Iterated Function Systems (IFSs) [Falconer], to produce fractals as limit sets of simple recursion mappings, and Topological Calculus [Arrighetti], which frames a topology-consistent, discrete counterpart to domains and operators [Giona]. Topological invariants and Analytical features of a set can be easily extracted form such a discrete model, even for complex geometries like fractal ones. The aim of this work is to show how recursion symmetries of a (pre-)\ fractal set, mathematically coded by ``algebraic'' relationships between its parts, are sole responsible for the self-similar distribution of its (laplacian) eigenvalues: no metric information is needed for this property to be observed. An example is given with the discrete version of a most known fractal set, the Serpinski gasket. Investigating the natural modes of self-similar domains is important to many applications whose core geometry is a prefractal itself (wires, drums, cavities, etc.). Most recently, transport [Giona] and electromagnetic pehnomena were focused: IFS-generated waveguides, resonators [Arrighetti] and antennas [Mosig] exhibiting multi-band properties. Such complex domains need careful mathematical tools, both in their continuum and discrete versions; requisite to them is the ability to take advantage from the topologic self-similarity, thus making simpler and well-known mathematical tools `available again' on the simplest geometries such fractals are formed by.