Non-hyperbolic iterated function systems: semifractals and the chaos game

Abstract

We consider iterated functions systems (IFS) on compact metric spaces and introduce the concept of target sets. Such sets have very rich dynamical properties and play a similar role as semifractals introduced by Lasota and Myjak do for regular IFSs. We study sufficient conditions which guarantee that the closure of the target set is a local attractor for the IFS. As a corollary, we establish necessary and sufficient conditions for the IFS having a global attractor. We give an example of a non-regular IFS whose target set is nonempty, showing that our approach gives rise to a "new class" of semifractals. Finally, we show that random orbits generated by IFSs draws target sets that are "stable".2000 Mathematics Subject Classification. 37C70, 28A80, 47H10.

Chaos in spatially homogeneous Hořava–Lifshitz subcritical cosmologies

“…a nonempty compact set A ⊆ K • such that lim n→∞ K n (C) = A, with respect to the Hausdorff metric for every nonempty compact set C ⊆ K • , see [22]. In case the maps are continuous, but the contractions are not uniform or weakly hyperbolic, there is still an attractor A when the diameter of iterates of the phase-space converge to zero for some sequence of symbols, see [1,29]. Note that in the usual physical time direction (i.e.…”

Visualization of the Chaos Game for non-hyperbolic iterated function system

“…Although theoretical progress has been made in the cases of non-hyperbolic IFS (M. Barnsley & Vince, 2011;Díaz & Matias, 2018;La Torre & Mendivil, 2013) this paper do not seek to present theoretical aspects that contribute to the works of these authors and theory in general, but it is sought to show the figures that result from applying the chaos game, with special emphasis on how functions and linear transformations associated with a non-hyperbolic IFS can create beautiful and complex but not necessarily fractal figures.…”