An Interpolation Scheme Involving Post Critically Finite Iterated Function Systems

We present an interpolation scheme that involves post critically finite iterated function systems. More precisely, while the classical fractal interpolation function, introduced by M. Barnsley, is the attractor of an iterated function system comprising Banach contractions on the metric space [x₀,xN]× K [ x 0 , x N ] × K (where N∈ N N ∈ N , N≥ 2 N ≥ 2 , \(xᵢ,yᵢ)∈ R² i∈ \0,1,...,N\\ { ( x i , y i ) ∈ R 2 ∣ i ∈ { 0 , 1 , . . . , N } } is a data set such that x₀<x₁<...<xN x 0 < x 1 < . . . < x N and K is a compact subset of R R ), with respect to the second variable, we enlarge the framework as follows: a) we replace [x₀,xN] [ x 0 , x N ] with a post critically finite iterated function system; b) we replace the Banach contraction condition with the Matkowski contraction condition, or the Meir-Keeler contraction condition, or the F-contraction condition, or the θ θ -contraction condition; c) in the case of Matkowski contraction condition, we replace K with R R . Moreover, our scheme includes zipper fractal interpolation functions.