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On Weakly Hyperbolic Iterated Function Systems

  • Alexander ArbietoEmail author
  • André Junqueira
  • Bruno Santiago
Article

Abstract

We study weakly hyperbolic iterated function systems on compact metric spaces, as defined by Edalat (Inform Comput 124(2):182–197, 1996), but in the more general setting of compact parameter space. We prove the existence of attractors, both in the topological and measure theoretical viewpoint and the ergodicity of invariant measure. We also define weakly hyperbolic iterated function systems for complete metric spaces and compact parameter space, extending the above mentioned definition. Furthermore, we study the question of existence of attractors in this setting. Finally, we prove a version of the results by Barnsley and Vince (Ergodic Theory Dyn Syst 31(4):1073–1079, 2011), about drawing the attractor (the so-called the chaos game), for compact parameter space.

Keywords

Iterated function systems Attractors Chaos game 

Mathematics Subject Classification

Primary 37B99 Secondary 37A05 

Notes

Acknowledgments

We would like to thank Katrin Gelfert and Daniel Oliveira for presenting us the paper of Kravchenko (2006).

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Copyright information

© Sociedade Brasileira de Matemática 2016

Authors and Affiliations

  • Alexander Arbieto
    • 1
    Email author
  • André Junqueira
    • 2
  • Bruno Santiago
    • 3
  1. 1.Universidade Federal do Rio de JaneiroRio de JaneiroBrazil
  2. 2.Universidade Federal de ViçosaViçosaBrazil
  3. 3.Université de BourgogneDijonFrance

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