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A Characterization of Self-similar Symbolic Spaces

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Abstract

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.

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Authors and Affiliations

  1. Area of Geometry and Topology, Faculty of Science, Universidad de Almería, La Cañada de San Urbano, 04120, Almería, Spain

    Francisco García Arenas & Miguel Angel Sánchez-Granero

Authors
  1. Francisco García Arenas
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  2. Miguel Angel Sánchez-Granero
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Corresponding author

Correspondence to Miguel Angel Sánchez-Granero.

Additional information

The second author acknowledges the support of the Spanish Ministry of Science and Innovation, grant MTM2009-12872-C02-01.

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Arenas, F.G., Sánchez-Granero, M.A. A Characterization of Self-similar Symbolic Spaces. Mediterr. J. Math. 9, 709–728 (2012). https://doi.org/10.1007/s00009-011-0146-4

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  • DOI: https://doi.org/10.1007/s00009-011-0146-4

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