iterated function systems

‎ ‎Let $(tilde{X},p)$ be the universal covering space of a compact metrizable space $ X $‎, ‎which is compact and locally path connected‎. ‎In this paper‎, ‎we show that there exist metrics $ d $ and $ d' $ for $ X $ and $tilde{X} $‎, ‎respectively‎, ‎such that any contractive map ‎$ f:Xto X $ induces a contractive map on $ tilde{X}$. ‎As an‎ application‎, ‎it is obtained that every iterated function system(IFS) on the space $ X $ with attractor $ K $‎, ‎induces an IFS on $tilde{ X} $ with attractor $tilde{K},$ such that $ p(tilde{K})=K$‎.

The aim of this paper is to construct a complete metric space of fuzzy valued image functions and to define a fractal transform operator T. Contraction of T is guarantees the existence of its fixed point. A fuzzy point is considered for this purpose as a crisp point and approached through classical method on proving the completeness of the space.

In this paper we consider shadowing and weak shadowing properties for iterated function systems IFS and give some results on these concepts. At first, a sufficient condition for shadowing property is given and by this result we present two IFS which have the shadowing property. It is proved that every uniformly expanding as well as every uniformly contracting IFS has the weak shadowing property. By an example we show that in IFS’s shadowing property does not imply weak shadowing property. Finally we have the main result of the paper and prove that the weak shadowing is a generic property in the set of all IFS’s.

In this paper, we introduce a new type of iterated function systems, named; CIFS. Actually in a CIFS we have some flows instead of some functions in iterated function systems. Then, we generalize the notions of average shadowing property, chain transitivity, and attractor sets on a CIFS. It is shown that every uniformly contracting CIFS has the average shadowing property. We also prove that if a CIFS, F on a compact metric space X has the average shadowing property, then F is chain transitive, but the converse is not always true. As a result, this proves that if F is an uniformly contracting CIFS on compact metric space X, then X is the only nonempty attractor of F.