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<p>One of the most important tools in the study of the dynamics of an iterated function system (IFS) is the canonical projection defined between the code space and the attractor of the system. The canonical projection is obtained as the fixed point of the <InlineEquation ID="IEq4"> <EquationSource Format="TEX">\(H_{S}\)</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>H</mi> <mi>S</mi> </msub> </math></EquationSource> </InlineEquation> operator associated with an IFS having a unique attractor. In this paper, we generalize the <InlineEquation ID="IEq5"> <EquationSource Format="TEX">\(H_{S}\)</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>H</mi> <mi>S</mi> </msub> </math></EquationSource> </InlineEquation> operator for a new class of IFSs for which the component functions are endowed with weaker contractivity conditions, so the attractor of such a system is not necessarily unique. We prove that the generalized operator is continuous and weakly Picard. Also, we use this generalization to prove that the Markov operator associated with such a system endowed with probabilities is weakly Picard.</p>

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A generalization of the \(H_{S}\) operator for pcIIFSs and oIIFSs

  • 期刊论文

  • 发表日期：

    2026年5月22日

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  • Alexandru Mihail,
  • Irina Savu

摘要

One of the most important tools in the study of the dynamics of an iterated function system (IFS) is the canonical projection defined between the code space and the attractor of the system. The canonical projection is obtained as the fixed point of the \(H_{S}\) H S operator associated with an IFS having a unique attractor. In this paper, we generalize the \(H_{S}\) H S operator for a new class of IFSs for which the component functions are endowed with weaker contractivity conditions, so the attractor of such a system is not necessarily unique. We prove that the generalized operator is continuous and weakly Picard. Also, we use this generalization to prove that the Markov operator associated with such a system endowed with probabilities is weakly Picard.