Blending attractors of Iterated Function Systems

TL;DR

This work develops a general, code-space–driven framework to blend fractal attractors from multiple contractive IFSs by forming a blending IFS on $K^*(X)$ and extracting blends $\mathcal{A}(\theta)$ as code-map images. It introduces a blending coefficient $\beta(\theta,i)$ to quantify similarity to each original attractor and proves continuity of blends with respect to the blending sequence. A rigorous discrete approximation method (BlendApprox) with explicit error bounds in the Hausdorff metric is provided, along with an algorithm to compute finite approximations within a prescribed tolerance. The paper demonstrates concrete blends (e.g., Sierpiński vs Maple leaf), applies Canright’s envelope for distance bounds, and extends the approach to invariant measures via IFSs with probabilities, highlighting the framework’s versatility and potential for further analysis and applications.

Abstract

In this paper we discuss a new method to blend fractal attractors using the code map for the IFS formed by the Hutchinson--Barnsley operators of a finite family of hyperbolic IFSs. We introduce a parameter called blending coefficient to measure the similarity between the blended set and each one of the original attractors. We also introduce a discrete approximation algorithm and prove a rigorous error estimation used to approximate these new objects. Several simulation results are provided illustrating our techniques.

Figures (6)

• Figure 1: Approximation of $A(\theta)$ via Algorithm BlendApprox.
• Figure 2: Sierpiński(left) and Maple(right).
• Figure 3: From left to the right, Sierpiński and Maple blends: (1) , (2) and (3).
• Figure 4: Attractor $R_3$.
• Figure 5: From left to the right, and up to bottom, blends (1) -- (8).

Theorems & Definitions (35)

• Definition 2.1
• Definition 2.2
• Definition 2.3
• Definition 2.4
• Theorem 2.5
• Definition 2.6
• Definition 2.7
• Theorem 2.8
• Theorem 2.8
• Definition 2.9