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

Aaryan Dharmesh Shah, Sangita Jha, Anarul Islam Mondal

TL;DR

This work develops a general nonlinear fractal interpolation framework on post-critically finite (PCF) self-similar sets using Edelstein contractions and nonconstant harmonic functions. It proves the existence and uniqueness of a graph-finite fractal interpolation function $f^*$ whose graph is the invariant set of a nonlinear IFS on $\mathcal{K}\times\mathcal{J}$ and satisfies $f^*(l_n(x)) = s_n(f^*(x)) + h_n(x)$ with prescribed data on $V_1$. The authors derive explicit box-dimension bounds for the graphs on the Sierpinski gasket and von-Koch curve, supported by examples, and analyze energy and normal derivatives, establishing finite-energy criteria in terms of $\Psi = \sum (L(s_n))^2$. By extending FIF theory beyond Lipschitz mappings, the paper provides a robust framework for nonlinear FIFs on a broad class of PCF fractals and clarifies their geometric and analytic properties.

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.

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

TL;DR

This work develops a general nonlinear fractal interpolation framework on post-critically finite (PCF) self-similar sets using Edelstein contractions and nonconstant harmonic functions. It proves the existence and uniqueness of a graph-finite fractal interpolation function whose graph is the invariant set of a nonlinear IFS on and satisfies with prescribed data on . The authors derive explicit box-dimension bounds for the graphs on the Sierpinski gasket and von-Koch curve, supported by examples, and analyze energy and normal derivatives, establishing finite-energy criteria in terms of . By extending FIF theory beyond Lipschitz mappings, the paper provides a robust framework for nonlinear FIFs on a broad class of PCF fractals and clarifies their geometric and analytic properties.

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.
Paper Structure (12 sections, 12 theorems, 117 equations, 7 figures)

This paper contains 12 sections, 12 theorems, 117 equations, 7 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Post Critically Finite Self-similar Structures
  4. Energy, Laplacian and Harmonic Function
  5. A Generalized Fractal Interpolation on PCF Self-similar Structures
  6. Dimensional Analysis on Certain Fractals
  7. Box-dimension of Non-linear FIF on SG
  8. Examples
  9. Box-dimension of Non-linear FIF on KC
  10. Properties of Uniform Non-Linear Fractal Functions
  11. Properties on SG
  12. Properties on KC

Key Result

Theorem 2.1

MeirKeeler Let $(X, d)$ be a complete metric space and $f: X \to X$ be a mapping satisfying Meircond. Then $f$ has a unique fixed point $\xi$. Moreover, for any $x \in X$,

Figures (7)

  • Figure 1: A non-linear FIF on the Sierpinski gasket
  • Figure 2: A non-linear FIF on the von-Koch curve
  • Figure 3: A non-linear FIF on SG with $\mu=2.75$
  • Figure 4: A non-linear FIF on SG with $\mu\leq1.8$
  • Figure 5: A non-linear FIF on KC with $\mu\leq2$
  • ...and 2 more figures

Theorems & Definitions (47)

  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Remark 2.1
  • Theorem 2.1
  • Definition 2.4
  • Definition 2.5
  • Definition 2.6
  • Definition 2.7
  • Definition 2.8
  • ...and 37 more