Deterministic and random iterated function systems

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Iterated function systems (abbreviated to IFSs) arise in many natural contexts. They are often used to encode and generate fractal images, such as landscapes and skyscapes, in computer games. They each generate, via a recursive procedure, a unique fractal set. They also play an important role in the theory of dynamical systems, and are in fact a generalisation of the process of looking at the backward trajectories of physical systems. My research aims at studying deterministic and random IFSs. Random IFSs are similar to their deterministic counterparts, but their generators are chosen randomly. The non-uniform outlook of the fractal sets they generate is usually more reminiscent of natural objects such as trees, clouds or mountains. In the 1980's and 1990's, pioneering work was accomplished on IFSs which comprise finitely many generators, each of these generators being a similarity map, that is, a transformation that preserves shapes. My research work concentrates on IFSs which are generated by finitely or even infinitely many transformations that do not necessarily preserve shapes, but that at least preserve angles. This type of IFSs has already found and will hopefully lead to new applications in number theory, complex dynamics and non-commutative algebra.