Iterated function systems and the code space

Abstract

The code space plays a significant role in the study of self-similar fractals. It is used to give coordinates to the points of a self-similar set. The code space also plays an important role in dynamical systems as well. The shift map on the code space is a valuable an example of a dynamical system. The shift map restricted to certain invariant subspaces is also important. The subshifts of finite type are used to analyze many common dynamical systems. This is the basis for the theory of symbolic dynamics. There is a particularly useful metric on the code space. With this metric the dimension of the code space and the subshifts of finite type can be computed using results of K. Falconer on sub-self-similar sets. We show that the code space can be embedded in Euclidean space, R n , by a map which is bi-Lipschitz. The shift or subshift of finite type on this embedded image can be extended to a C ∞-map F : R n→ R n having the embedded set as a maximal compact invariant set which contains the nonwandering set of F in this case. The map F restricted to this set will be equivalent to a power of the shift or the subshift of finite type. The Hausdorff dimension of the invariant set, the Lyapunov exponents of F at various points in the invariant set, and the topological entropy of F can all be computed. This provides a general method of constructing examples in which the relationship between these quantities can be studied.