On the Complexity of some Geometrical Objects

We recall the definition of the $ϵ$-distortion complexity of a set defined in \cite{bcc} and the results obtained in this paper for Cantor sets of the interval defined by iterated function systems. We state an analogous definition for measures which may be more useful when dealing with dynamical systems. We prove a new lower bound in the case of Cantor sets of the interval defined by analytic iterated function systems. We also give an upper bound the $ϵ$-distortion complexity of invariant sets of uniformly hyperbolic dynamical systems.