Analysis of non-linear fractal functions on PCF self-similar sets

TL;DR

This paper introduces a general framework for constructing non-linear fractal functions on PCF self-similar sets, analyzing their properties, and estimating their box dimensions, with applications to classic fractals like the Sierpinski gasket.

Contribution

It develops a broad method for creating non-linear fractal functions on PCF sets using Edelstein contractions, expanding the class of such functions and analyzing their dimensions.

Findings

Constructed non-linear fractal interpolation functions on PCF sets.

Calculated upper and lower box dimensions of fractal graphs.

Provided numerical examples illustrating the theoretical results.

Abstract

This article deals with (1) the construction of a general non-linear fractal interpolation function on PCF self-similar sets, (2) the energy and normal derivatives of uniform non-linear fractal functions, (3) estimation of the bound of box dimension of the proposed fractal functions on the Sierpinski gasket and the von-Koch curve. Here, we present a more general framework to construct the attractor and the functions on the PCF self-similar sets using the Edelstein contraction, which broadens the class of functions. En route, we calculate the upper and lower box dimensions of the graph of non-linear interpolant. Finally, we provide several graphical and numerical examples for illustration of the construction and estimate the dimensions for different data sets.