A Characterization of Self-similar Symbolic Spaces

In this paper we use fractal structures to study self-similar sets and self-similar symbolic spaces. We show that these spaces have a natural fractal structure, justifying the name of fractal structure, and we characterize self-similar symbolic spaces in terms of fractal structures. We prove that self-similar symbolic spaces can be characterized in a similar way, in the form, to the definition of classical self-similar sets by means of iterated function systems. We also study when a self-similar symbolic space is a self-similar set. Finally, we study relations between fractal structures with “pieces” homeomorphic to the space and different concepts of self-homeomorphic spaces. Along the paper, we propose several methods in order to construct self-similar sets and self-similar symbolic spaces from a geometrical approach. This allows to construct these kind of spaces in a very easy way.