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Study of attractors and fractal functions on the product spaces and Dimensional aspects

Alamgir Hossain

Abstract

In this paper, the product of the Hausdorff metric on the product space is defined and the equivalency between the product Hausdorff metric and the Hausdorff metric on the product space is established. The finite product of the iterated function systems (IFS) on the product space is considered and the relation between the attractor of the product IFS and the attractors of the co-ordinate IFSs is studied. Dimension bounds of the homogeneous and inhomogeneous attractors on the product space is established. Also, the product fractal interpolation function on the higher dimensional space is constructed.

Study of attractors and fractal functions on the product spaces and Dimensional aspects

Abstract

In this paper, the product of the Hausdorff metric on the product space is defined and the equivalency between the product Hausdorff metric and the Hausdorff metric on the product space is established. The finite product of the iterated function systems (IFS) on the product space is considered and the relation between the attractor of the product IFS and the attractors of the co-ordinate IFSs is studied. Dimension bounds of the homogeneous and inhomogeneous attractors on the product space is established. Also, the product fractal interpolation function on the higher dimensional space is constructed.
Paper Structure (14 sections, 28 theorems, 125 equations, 4 figures)

This paper contains 14 sections, 28 theorems, 125 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Some notations and preliminary results
  3. Product metric
  4. Cartesian product of the functions
  5. Hausdorff metric
  6. Homogeneous attractor
  7. Inhomogeneous attractor
  8. Relation between homogeneous and inhomogeneous attractors
  9. Fractal function
  10. Hausdorff, packing and box dimensions
  11. Dimension of the product set
  12. Homogeneous attractor on the product space and dimension bounds
  13. Inhomogeneous attractor on the product spaces and dimension bounds
  14. Construction of fractal function on the product space

Key Result

Lemma 2.1

FE If $\left\{C_i\right\}_{i\in\Lambda}$, $\left\{D_i\right\}_{i\in\Lambda}$ are two finite collections of sets in $\left ( \mathscr{H}(Y), h \right)$, then

Figures (4)

  • Figure 1: First three levels in the construction of Sierpinski triangle with condensation set.
  • Figure 2: The product $F_1\times F_2$.
  • Figure 3: The product $F\times F$, where $F$ is the middle third Cantor set.
  • Figure 4: First three levels in the construction of inhomogeneous product attractor with condensation set.

Theorems & Definitions (56)

  • Lemma 2.1
  • Theorem 2.1
  • Definition 2.1
  • Theorem 2.2
  • Definition 2.2
  • Definition 2.3
  • Theorem 2.3
  • Example 2.1
  • Lemma 2.2
  • proof
  • ...and 46 more