• Title/Summary/Keyword: infinite iterated function system

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QUANTIZATION FOR A PROBABILITY DISTRIBUTION GENERATED BY AN INFINITE ITERATED FUNCTION SYSTEM

  • Roychowdhury, Lakshmi;Roychowdhury, Mrinal Kanti
    • Communications of the Korean Mathematical Society
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    • v.37 no.3
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    • pp.765-800
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    • 2022
  • Quantization for probability distributions concerns the best approximation of a d-dimensional probability distribution P by a discrete probability with a given number n of supporting points. In this paper, we have considered a probability measure generated by an infinite iterated function system associated with a probability vector on ℝ. For such a probability measure P, an induction formula to determine the optimal sets of n-means and the nth quantization error for every natural number n is given. In addition, using the induction formula we give some results and observations about the optimal sets of n-means for all n ≥ 2.

ON TOPOLOGICAL ENTROPY AND TOPOLOGICAL PRESSURE OF NON-AUTONOMOUS ITERATED FUNCTION SYSTEMS

  • Ghane, Fatemeh H.;Sarkooh, Javad Nazarian
    • Journal of the Korean Mathematical Society
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    • v.56 no.6
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    • pp.1561-1597
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    • 2019
  • In this paper we introduce the notions of topological entropy and topological pressure for non-autonomous iterated function systems (or NAIFSs for short) on countably infinite alphabets. NAIFSs differ from the usual (autonomous) iterated function systems, they are given [32] by a sequence of collections of continuous maps on a compact topological space, where maps are allowed to vary between iterations. Several basic properties of topological pressure and topological entropy of NAIFSs are provided. Especially, we generalize the classical Bowen's result to NAIFSs ensures that the topological entropy is concentrated on the set of nonwandering points. Then, we define the notion of specification property, under which, the NAIFSs have positive topological entropy and all points are entropy points. In particular, each NAIFS with the specification property is topologically chaotic. Additionally, the ${\ast}$-expansive property for NAIFSs is introduced. We will prove that the topological pressure of any continuous potential can be computed as a limit at a definite size scale whenever the NAIFS satisfies the ${\ast}$-expansive property. Finally, we study the NAIFSs induced by expanding maps. We prove that these NAIFSs having the specification and ${\ast}$-expansive properties.