Fractal Riemann Surfaces: Chaotic Scenarios and Applications

Fractal Riemann surfaces are generated as iterators of branched covers (complex multi-valued functions). They feature self-similar geometries, an interesting iterated monodromy group (IMG) driving their topologies, and an easy way to get their symbolic dynamics browsed. On the contrary, convergence issues, numerical accuracy and the onset of chaotic dynamics are present in the direct, homotopy problem of computing paths on them. Theoretical results of analysis and synthesis will be given, with a final look to possible applications in Computer Science (signing and private-key cryptography) and Physics (scattering in fractal resonators).