Elementary fractal geometry. 6. The dynamical interior of self-similar sets

Abstract

On the one hand, the dynamical interior of a self-similar set with open set condition is the complement of the dynamical boundary. On the other hand, the dynamical interior is the recurrent set of the magnification flow. For a finite type self-similar set, both boundary and interior are described by finite automata. The neighbor graph defines the boundary. The neighborhood graph, based on work by Thurston, Lalley, Ngai and Wang, defines the interior. If local views are considered up to similarity, the interior obtains a discrete manifold structure, and the magnification flow is discretized by a Markov chain. This leads to new methods for the visualization and description of finite type attractors.