Numerics and Fractals

TL;DR

This paper explores local iterated function systems, their fixed points as local fractal functions, and introduces methods for their computation, including special cases like piecewise polynomials and data-driven approaches.

Contribution

It establishes existence and properties of local fractal functions, linking them to piecewise polynomials and developing a data-based computation method.

Findings

Local fractal functions include piecewise polynomials as special cases.

Methods for computing local IFS components from data or PDEs are developed.

The paper discusses properties and fixed points of local fractal functions.

Abstract

Local iterated function systems are an important generalisation of the standard (global) iterated function systems (IFSs). For a particular class of mappings, their fixed points are the graphs of local fractal functions and these functions themselves are known to be the fixed points of an associated Read-Bajactarevi\'c operator. This paper establishes existence and properties of local fractal functions and discusses how they are computed. In particular, it is shown that piecewise polynomials are a special case of local fractal functions. Finally, we develop a method to compute the components of a local IFS from data or (partial differential) equations.