On Weakly Hyperbolic Iterated Function Systems

Alexander Arbieto (1) , André Junqueira (2) , Bruno Santiago (3)

1 UFRJ - Universidade Federal do Rio de Janeiro [Brasil] = Federal University of Rio de Janeiro [Brazil] = Université fédérale de Rio de Janeiro [Brésil]
2 UFV - Universidade Federal de Viçosa = Federal University of Viçosa
3 IMB - Institut de Mathématiques de Bourgogne [Dijon]

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.

Mots clés: Iterated function systems; Attractors; Chaos game

Domaines: Mathématiques [math]

Alexander Arbieto, André Junqueira, Bruno Santiago. On Weakly Hyperbolic Iterated Function Systems. Boletim da Sociedade Brasileira de Matemática / Bulletin of the Brazilian Mathematical Society, 2017, 48 (1), pp.111 - 140. ⟨10.1007/s00574-016-0018-4⟩. ⟨hal-01583500⟩