Elementary fractal geometry. 4. Automata-generated topological spaces

Abstract

Finite automata were used to determine multiple addresses in number systems and to find topological properties of self-affine tiles and finite type fractals. We join these two lines of research by axiomatically defining automata which generate topological spaces. Simple examples show the potential of the concept. Spaces generated by automata are topologically self-similar. Two basic algorithms are outlined. The first one determines automata for all $k$-tuples of equivalent addresses from the automaton for double addresses. The second one constructs finite topological spaces which approximate the generated space. Finally, we discuss the realization of automata-generated spaces as self-similar sets.