Some Remarks on Fractals Generated by a Sequence of Finite Systems of Contractions
Abstract We generalize some results shown by J. E. Hutchinson in [Indiana Univ. Math. J. 30: 713–747, 1981]. Let be finite systems of contractions on a complete metric space; then, under some conditions on (𝔉𝑛), there exists a unique non-empty compact set 𝐾 such that the sequence defined by ((𝔉1 ○ 𝔉2 ○ ⋯ ○ 𝔉𝑛)(𝐶)) converges to 𝐾 in the Hausdorff metric for every non-empty closed and bounded set 𝐶. If the metric space is also separable and for every there are real numbers strictly between 0 and 1, satisfying the condition , then there exists a unique probability Radon measure μ 𝐾 such that the sequence weakly converges to μ 𝐾 for every probability Borel regular measure ν with bounded support (where by we denote the image measure of ν under a contraction 𝑓). Moreover, 𝐾 is the support of μ 𝐾.