On the chaos game of iterated function systems. (English) Zbl 1373.37054

Topol. Methods Nonlinear Anal. 49, No. 1, 105-132 (2017); addendum and corrigendum ibid. 55, No. 2, 601-616 (2020).
Summary: Every quasi-attractor of an iterated function system (IFS) of continuous functions on a first-countable Hausdorff topological space is renderable by the probabilistic chaos game. By contrast, we prove that the backward minimality is a necessary condition to get the deterministic chaos game. As a consequence, we obtain that an IFS of homeomorphisms of the circle is renderable by the deterministic chaos game if and only if it is forward and backward minimal. This result provides examples of attractors (a forward but no backward minimal IFS on the circle) that are not renderable by the deterministic chaos game. We also prove that every well-fibred quasi-attractor is renderable by the deterministic chaos game as well as quasi-attractors of both, symmetric and non-expansive IFSs.


37C05 Dynamical systems involving smooth mappings and diffeomorphisms
37C20 Generic properties, structural stability of dynamical systems
37E30 Dynamical systems involving homeomorphisms and diffeomorphisms of planes and surfaces
37D45 Strange attractors, chaotic dynamics of systems with hyperbolic behavior
37C70 Attractors and repellers of smooth dynamical systems and their topological structure
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