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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 introduces the notion of trajectories of maps defined by function systems which may be considered as a new generalization of the traditional IFS. The significance and the convergence properties of forward and backward trajectories are studied. In contrast to the ordinary fractals which are self-similar at different scales, the attractors of these trajectories may have different structures at different scales.
We study the topological properties of attractors of Iterated Function Systems (I.F.S.) on the real line, consisting of affine maps of homogeneous contraction ratio. These maps define what we call a second generation I.F.S.: they are uncountably many
The conservative sequence of a set $A$ under a transformation $T$ is the set of all $n in mathbb{Z}$ such that $T^n A cap A ot = varnothing$. By studying these sequences, we prove that given any countable collection of nonsingular transformations wi
Attractors of cooperative dynamical systems are particularly simple; for example, a nontrivial periodic orbit cannot be an attractor. This paper provides characterizations of attractors for the wider class of coherent systems, defined by the property
Spatio-temporal pattern formation over the square and rectangular domain has received significant attention from researchers. A wide range of stationary and non-stationary patterns produced by two interacting populations is abundant in the literature
This paper shows that the celebrated Embedding Theorem of Takens is a particular case of a much more general statement according to which, randomly generated linear state-space representations of generic observations of an invertible dynamical system