JournalsjfgVol. 1 , No. 3pp. 243–271

A topological separation condition for fractal attractors

  • Tim Bedford

    University of Strathclyde, Glasgow, Great Britain
  • Sergiy V. Borodachov

    Towson University, USA
  • Jeffrey S. Geronimo

    Georgia Institute of Technology, Atlanta, USA
A topological separation condition for fractal attractors cover
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Abstract

We consider finite systems of contractive homeomorphisms of a complete metric space, which satisfy the minimality property. In general this separation condition is weaker than the strong open set condition and is not equivalent to the weak separation property. We prove that this separation condition is equivalent to the strong Markov property (see definition below). We also show that the set of NN-tuples of contractive homeomorphisms, having the minimality property, is a GδG_\delta set in the topology of pointwise convergence of every component mapping with an additional requirement that the supremum of contraction coefficients of mappings in the sequence be strictly less than one. We find a class of NN-tuples of d×dd\times d invertible contraction matrices, which define systems of affine mappings in Rd\mathbb R^d having the minimality property for almost every NN-tuple of fixed points with respect to the NdNd-dimensional Lebesgue measure.

Cite this article

Tim Bedford, Sergiy V. Borodachov, Jeffrey S. Geronimo, A topological separation condition for fractal attractors. J. Fractal Geom. 1 (2014), no. 3 pp. 243–271

DOI 10.4171/JFG/7