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<p>The classical Hutchinson–Barnsley theory, foundational to hyperbolic iterated function systems (IFSs), embodies a rich interplay between fractal geometry and fixed-point theory, focusing on the existence and uniqueness of a fractal set and a fractal measure. Originating from Hutchinson’s seminal paper (Indiana Univ Math J 30:713–747, 1981), this theory has undergone numerous extensions that explore different types of IFS. These extensions investigate the fractal set and fractal measure by employing suitable fixed-point theorems applied to the Hutchinson–Barnsley operator and Markov operator associated with the IFS. In the pursuit of an axiomatic theory for IFS, it has been noted that favorable properties of the code map linked to an IFS facilitate deriving key conclusions of the classical Hutchinson–Barnsley theory. In this context, this note delves into the properties of the code map associated with an IFS composed of local radial contractions. Leveraging these established properties, the existence of a unique invariant set and a unique invariant measure for the IFS is deduced, circumventing the need to rely on the properties of the hyperspace of compact subsets and the Hausdorff metric.</p>

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The point-fibred property of an iterated function system with local radial contractions and its consequences

  • 期刊论文

  • 发表日期：

    2025年2月25日

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  • R. Medhi,
  • P. Viswanathan

摘要

The classical Hutchinson–Barnsley theory, foundational to hyperbolic iterated function systems (IFSs), embodies a rich interplay between fractal geometry and fixed-point theory, focusing on the existence and uniqueness of a fractal set and a fractal measure. Originating from Hutchinson’s seminal paper (Indiana Univ Math J 30:713–747, 1981), this theory has undergone numerous extensions that explore different types of IFS. These extensions investigate the fractal set and fractal measure by employing suitable fixed-point theorems applied to the Hutchinson–Barnsley operator and Markov operator associated with the IFS. In the pursuit of an axiomatic theory for IFS, it has been noted that favorable properties of the code map linked to an IFS facilitate deriving key conclusions of the classical Hutchinson–Barnsley theory. In this context, this note delves into the properties of the code map associated with an IFS composed of local radial contractions. Leveraging these established properties, the existence of a unique invariant set and a unique invariant measure for the IFS is deduced, circumventing the need to rely on the properties of the hyperspace of compact subsets and the Hausdorff metric.