Some Recent Progress Concerning Topology of Fractals

We give a panorama of recent results in fractal geometry, concerning properties of attractors of iterated function systems (IFS). The work we present has a topological flavour but is motivated mainly by geometry. We review the various notions of attractor, then introduce the symbolic dynamics language for handling IFS and review its application to the exciting new area of fractal homeomorphisms. We also recall Kameyama’s question on topological contractivity of reasonable IFSs, and discuss IFS theory from the point of view of Conley decompositions. We review the recently discovered theory of projective IFSs, which live in beautiful environments, projective spaces, with their own specific structures that generalize affine IFS theory. We conclude by explaining the random iteration algorithm from a topological point of view. This algorithm, which often reveals attractors, works due to an intimate connection between the tree structure of code space and the stochastic process that generates the code. The result is a deterministic version of the "chaos game" which always works and avoids probabilistic notions.