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Non-stationary versions of fixed-point theory, with applications to fractals and subdivision

  • David LevinEmail author
  • Nira Dyn
  • Viswanathan Puthan Veedu
Article
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Abstract

Iterated Function Systems (IFSs) have been at the heart of fractal geometry almost from its origin, and several generalizations for the notion of IFS have been suggested. Subdivision schemes are widely used in computer graphics and attempts have been made to link fractals generated by IFSs to limits generated by subdivision schemes. With an eye towards establishing connection between non-stationary subdivision schemes and fractals, this paper investigates “trajectories of maps defined by function systems” which are considered as generalizations of the traditional IFS. The significance of ’forward’ and ’backward’ trajectories of general sequences of maps is studied. The convergence properties of these trajectories constitute a non-stationary version of the classical fixed-point theory. Unlike the ordinary fractals which are self-similar at different scales, the attractors of trajectories of maps defined by function systems may have different structures at different scales.

Keywords

Fractals subdivision schemes fixed-point theory attractors 

Mathematics Subject Classification

28A80 47H10 54E50 41A30 

Notes

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Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  1. 1.School of Mathematical SciencesTel Aviv UniversityTel AvivIsrael
  2. 2.Department of MathematicsIndian Institute of Technology DelhiNew DelhiIndia

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