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The discrete charm of iterated function systems. A computer scientist's perspective on approximation of IFS invariant sets and measures

Tomasz Martyn

TL;DR

This work investigates the possibility of the application of the random iteration algorithm to approximate discrete IFS invariant sets and measures generated by iterated function systems defined on countable discrete spaces that are uniform grids of a finite dimension.

Abstract

We study invariant sets and measures generated by iterated function systems defined on countable discrete spaces that are uniform grids of a finite dimension. The discrete spaces of this type can be considered as models of spaces in which actual numerical computation takes place. In this context, we investigate the possibility of the application of the random iteration algorithm to approximate these discrete IFS invariant sets and measures. The problems concerning a discretization of hyperbolic IFSs are considered as special cases of this more general setting.

The discrete charm of iterated function systems. A computer scientist's perspective on approximation of IFS invariant sets and measures

TL;DR

This work investigates the possibility of the application of the random iteration algorithm to approximate discrete IFS invariant sets and measures generated by iterated function systems defined on countable discrete spaces that are uniform grids of a finite dimension.

Abstract

We study invariant sets and measures generated by iterated function systems defined on countable discrete spaces that are uniform grids of a finite dimension. The discrete spaces of this type can be considered as models of spaces in which actual numerical computation takes place. In this context, we investigate the possibility of the application of the random iteration algorithm to approximate these discrete IFS invariant sets and measures. The problems concerning a discretization of hyperbolic IFSs are considered as special cases of this more general setting.

Paper Structure

This paper contains 11 sections, 29 theorems, 95 equations, 5 figures.

Table of Contents

  1. The problem
  2. Previous work
  3. Organization of the paper
  4. Minimal absorbing sets
  5. ...more often than rarely
  6. Discrete Iterated Function System
  7. On the existence of multiple attractors of a discretized hyperbolic IFS in practice
  8. Invariant measure and stationary distribution
  9. To be attractive (or at least ergodic)
  10. DIFSs for hyperbolic IFSs
  11. Open problems

Key Result

Theorem 4.1

For $\tilde{w} : \mathcal{D}^{n}(\delta) \rightarrow \mathcal{D}^{n}(\delta)$ and any nonempty set $C \supset \tilde{w}(C)$ the following statements hold: (a) Every absorbing set is nonempty (by definition). (b) Since $C$ is nonempty, then by definition $C$ is an absorbing set in $C$, so there is al

Figures (5)

  • Figure 1: Examples of minimal absorbing sets and their basins of attractions generated by $\delta$-roundoffs of two-dimensional linear contractions. From left to right, top to bottom, the first five pictures are derived from similarities with scaling factors and rotations respectively: $0.6$ and $0^{\circ}$, $0.6$ and $5^{\circ}$, $0.6$ and $150^{\circ}$, $0.6$ and $30^{\circ}$, $0.9$ and $30^{\circ}$. The last picture is derived from a linear mapping specified by the matrix $[0.50.3-0.10.4]$.
  • Figure 2: The basins of attractions generated by $\delta$-roundoffs of contractive similarities with the linear part specified by scaling factor $0.45$ and rotation $15^{\circ}$, and fixed points in $[-0.4, 0.4]^2$
  • Figure 3: Statistics related to a minimal absorbing set of a $\delta$-roundoff of a 2D affine contraction with respect to the contractivity factor less than a given value (see the main text): (a) the probability for the minimal absorbing set generated by an affine (blue) and similarity (red) contraction to be a non-singleton and to have more than one component (green and orange plots, respectively); (b) the expected value of the number of the a minimal absorbing set components
  • Figure 4: Same as in Fig. \ref{['fig:3']}, but with respect to the $d_{\infty}$ distance of the fixed point of a mapping to the center of the $\delta$-cube (see the main text). In (b) additionally linear trends are shown
  • Figure 5: Examples of DIFS stationary distributions visualized with RIA. The DIFSs are composed of three maps. The number of the minimal absorbing set components of the maps are: a) 2, 3, 6, b) 1, 13, 1, c) 1, 11, 5. The basins of attractions of the maps are depicted on the left-hand side of the pictures

Theorems & Definitions (60)

  • Definition 4.1
  • Definition 4.2
  • Definition 4.3
  • Theorem 4.1
  • Theorem 4.2
  • proof
  • Remark 4.1
  • Definition 4.4
  • Corollary 4.3
  • Theorem 4.4
  • ...and 50 more