Attractors of trees of maps and of sequences of maps between spaces with applications to subdivision

  • Nira Dyn
  • David Levin
  • Peter MassopustEmail author


An extension of the Banach fixed-point theorem for a sequence of maps on a complete metric space (Xd) has been presented in a previous paper. It has been shown that backward trajectories of maps \(X\rightarrow X\) converge under mild conditions and that they can generate new types of attractors such as scale-dependent fractals. Here we present two generalizations of this result and some potential applications. First, we study the structure of an infinite tree of maps \(X\rightarrow X\) and discuss convergence to a unique “attractor” of the tree. We also consider “staircase” sequences of maps, that is, we consider a countable sequence of metric spaces \(\{(X_i,d_i)\}\) and an associated countable sequence of maps \(\{T_i\}\), \(T_i:X_{i}\rightarrow X_{i-1}\). We examine conditions for the convergence of backward trajectories of the \(\{T_i\}\) to a unique attractor. An example of such trees of maps are trees of function systems leading to the construction of fractals which are both scale-dependent and location dependent. The staircase structure facilitates linking all types of linear subdivision schemes to attractors of function systems.


Fractals subdivision schemes fixed points attractors function systems 

Mathematics Subject Classification

Primary 47H10 Secondary 28A80 41A30 54E50 



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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.Centre of MathematicsTechnical University of MunichMunichGermany

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