2016/05/09 by Edgar Matias, Matias, Edgar, Lorenzo J. Díaz +1
Mathematics · Physics and Astronomy · #37B25 #37B35 #47B80 #60J05 #Dynamical Systems (math.DS) #FOS: Mathematics #Mathematical Dynamics and Fractals #Quantum chaos and dynamical systems
paper · pdf · doi:10.48550/arxiv.1605.02752
openalex publication_date 2016/05/09 · openalex created_date 2022/09/12 · openalex updated_date 2026/07/28
We consider iterated function systems \IFS(T1,\…,Tk)\nconsisting of continuous self maps of a compact metric space X. We introduce\nthe subset S\t of \weakly hyperbolic sequences\n\ξ=\ξ0\…\ξn \… \∈ \Σk+ having the property that\n bigcapn T_\ξ0\∘\⋯\∘ T_\ξn(X) is a point\n \π(\ξ) . The target set \π(S\t) plays a role similar to\nthe semifractal introduced by Lasota-Myjak.\n Assuming that S\t\≠ \∅ (the only hyperbolic-like\ncondition we assume) we prove that the IFS has at most one strict attractor and\nwe state a sufficient condition guaranteeing that the strict attractor is the\nclosure of the target set. Our approach applies to a large class of genuinely\nnon-hyperbolic IFSs (e.g. with maps with expanding fixed points) and provides a\nnecessary and sufficient condition for the existence of a globally attracting\nfixed point of the Barnsley-Hutchinson operator. We provide sufficient\nconditions under which the disjunctive chaos game yields the target set (even\nwhen it is not a strict attractor).\n We state a sufficient condition for the asymptotic stability of the Markov\noperator of a recurrent IFS. For IFSs defined on [0,1] we give a simple\ncondition for their asymptotic stability. In the particular case of IFSs with\nprobabilities satisfying a "locally injectivity" condition, we prove that if\nthe target set has at least two elements then the Markov operator is\nasymptotically stable and its stationary measure is supported in the closure of\nthe target set.\n