9 - Further Examples: Attractors and Limit Sets

Summary

In this chapter we collect together discussion of several further families of fractal set. In Section 9.1 we consider self-conformal sets, which are another special case of IFS attractor. In Section 9.2 we consider sets invariant under parabolic interval maps. While these sets are similar in spirit to IFS attractors, they are not necessarily attractors of IFSs since the inverse branches of the associated dynamical system fail to be strict contractions. The resulting parabolic behaviour greatly influences the Assouad dimension of such sets. In Section 9.3 we consider limit sets of Kleinian groups which are invariant sets for the group action on the boundary of hyperbolic space.In Section 9.4 we consider the random limit sets resulting from Mandelbrot percolation.