Topological Calculus: between Algebraic Topology and Electromagnetic Fields

Topological behaviour of self-similar spectra for fractal domains is shown and applied to solve electromagnetic problems on fractal geometries, like for example the Sierpinski gasket.. Two different mathematical tools are employed: the Topological Calculus, which frames a topology-consistent, discrete counterpart to domains and operators and the Iterated Function Systems (IFSs) to produce fractals as limit sets of simple recursion mappings. Topological invariants and Analytical features of a set can be easily extracted from such a discrete model, even for complex geometries like fractal ones.One of the targets 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.Another primary target is to show how Topological Calculus easily allows for an almost instantaneous discretization of contoinuum equations of any (topological) field theory. Investigating the natural modes of self-similar domains is important to many applications whose core geometry is prefractal or at least highly irregular. Most recently, transport and electromagnetic pehnomena were focused: IFS-generated waveguides, resonators and antennas(9) exhibiting multi-band properties. Such complex domains need careful mathematical formulations in order to transfer traditional-geometric properties to them; Topological Calculus is one of such discrete formulations.