JournalsjfgVol. 3 , No. 3pp. 217–243

Conformal graph directed Markov systems: beyond finite irreducibility

  • Andrei E. Ghenciu

    University of Wisconsin-Stout, Menomonie, USA
  • R. Daniel Mauldin

    University of North Texas, Denton, USA
  • Mario Roy

    York University, Toronto, Canada
Conformal graph directed Markov systems: beyond finite irreducibility cover
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Abstract

In this paper, we study CGDMSs which are not necessarily finitely irreducible. We do this from two different perspectives: by investigating irreducible infinite systems and by examining general (i.e. potentially reducible) finite systems. In this latter case, we derive a necessary and sufficient condition under which the Hausdorff measure of the limit set is positive and finite. We further show that if this condition doesn’t hold then the Hausdorff measure, though infinite, is σ\sigma-finite. This condition is given in terms of the strongly connected components of the directed graph. We also demonstrate that the pressure and Hausdorff dimension of the limit set are completely determined by the strongly connected components.

Cite this article

Andrei E. Ghenciu, R. Daniel Mauldin, Mario Roy, Conformal graph directed Markov systems: beyond finite irreducibility. J. Fractal Geom. 3 (2016), no. 3 pp. 217–243

DOI 10.4171/JFG/35