Developments in fractal geometry

Abstract

Iterated function systems have been at the heart of fractal geometry almost from its origins. The purpose of this expository article is to discuss new research trends that are at the core of the theory of iterated function systems (IFSs). The focus is on geometrically simple systems with finitely many maps, such as affine, projective and Möbius IFSs. There is an emphasis on topological and dynamical systems aspects. Particular topics include the role of contractive functions on the existence of an attractor (of an IFS), chaos game orbits for approximating an attractor, a phase transition to an attractor depending on the joint spectral radius, the classification of attractors according to fibres and according to overlap, the kneading invariant of an attractor, the Mandelbrot set of a family of IFSs, fractal transformations between pairs of attractors, tilings by copies of an attractor, a generalization of analytic continuation to fractal functions, and attractor-repeller pairs and the Conley "landscape picture" for an IFS.

Model-independent parametrization of the hadronic vacuum polarization andg-2 for the muon on the lattice

“…These "standard" multi-point PAs were studied in Refs. [12,13]. By construction, they are exact at the values Q 2 = Q 2 i , i.e., the PA takes precisely the values Φ(Q 2 i ) at these values of Q 2 .…”

Fast Basins and Branched Fractal Manifolds of Attractors of Iterated Function Systems

“…An IFS may not possess an attractor, or it may possess multiple attractors. Examples and further discussion can be found in [5] and in references therein.…”

“…So there is an infinite subsequence {γ n } of {γ n } and the corresponding subsequence {β n } of {β n } such that γ n ∈ p(α, β n ) = p(α, β 0 ) for some fixed β 0 . Since α, γ n ∈ p(α) for all n and p(α) is compact by statement (5), it follows from the fact that π(γ n ) converges to π(α) and statement (4) that {γ n } converges to α. But from the facts that p(α, β 0 ) is compact and γ n ∈ p(α, β 0 ), it follows that α ∈ p(α, β 0 ).…”

Implementasi Teorema Collage Untuk Mendesain Sistem Fungsi Iterasi (Sfi): Studi Awal Mendesain Sfi Motif Songket Lombok

“…Himpunan Mandelbrot untuk berbagai skala [2] Sistem Fungsi Iterasi (SFI) merupakan teknik yang dapat digunakan untuk mengkonstruksi model-model fraktal. Menurut Barnsley, SFI dipandang sebagai inti geometri fraktal [6]. Sebagai contoh, SFI sierpinski triangle pada Gambar 2 diberikan oleh himpunan pemetaan kontraktif:…”