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On the Higuchi fractal dimension of invariant measures for countable idempotent iterated function systems

Elismar R. Oliveira

TL;DR

This work tackles invariant idempotent measures for countable max-plus IFS in compact spaces, proving existence and uniqueness under constant weights and showing that partial-attractor invariants converge to the full system. It bridges to fuzzy dynamics via a density-based mapping, establishing a correspondence between idempotent invariant measures and fuzzy fixed points, and proves contraction properties that justify finite-precision approximations. Numerically, the authors discretize partial attractors and apply Higuchi-type fractal analysis to the resulting fuzzy representations, revealing increasing fractal complexity as the system is refined in both 1-D and 2-D settings. The results provide a concrete, quantifiable framework for approximating invariant idempotent measures and for characterizing their geometric complexity, with potential applications to discrete data and gray-level image analysis.

Abstract

We investigate the set of invariant idempotent probabilities for countable idempotent iterated function systems (IFS) defined in compact metric spaces. We demonstrate that, with constant weights, there exists a unique invariant idempotent probability. Utilizing Secelean's approach to countable IFSs, we introduce partially finite idempotent IFSs and prove that the sequence of invariant idempotent measures for these systems converges to the invariant measure of the original countable IFS. We then apply these results to approximate such measures with discrete systems, producing, in the one-dimensional case, data series whose Higuchi fractal dimension can be calculated. Finally, we provide numerical approximations for two-dimensional cases and discuss the application of generalized Higuchi dimensions in these scenarios.

On the Higuchi fractal dimension of invariant measures for countable idempotent iterated function systems

TL;DR

This work tackles invariant idempotent measures for countable max-plus IFS in compact spaces, proving existence and uniqueness under constant weights and showing that partial-attractor invariants converge to the full system. It bridges to fuzzy dynamics via a density-based mapping, establishing a correspondence between idempotent invariant measures and fuzzy fixed points, and proves contraction properties that justify finite-precision approximations. Numerically, the authors discretize partial attractors and apply Higuchi-type fractal analysis to the resulting fuzzy representations, revealing increasing fractal complexity as the system is refined in both 1-D and 2-D settings. The results provide a concrete, quantifiable framework for approximating invariant idempotent measures and for characterizing their geometric complexity, with potential applications to discrete data and gray-level image analysis.

Abstract

We investigate the set of invariant idempotent probabilities for countable idempotent iterated function systems (IFS) defined in compact metric spaces. We demonstrate that, with constant weights, there exists a unique invariant idempotent probability. Utilizing Secelean's approach to countable IFSs, we introduce partially finite idempotent IFSs and prove that the sequence of invariant idempotent measures for these systems converges to the invariant measure of the original countable IFS. We then apply these results to approximate such measures with discrete systems, producing, in the one-dimensional case, data series whose Higuchi fractal dimension can be calculated. Finally, we provide numerical approximations for two-dimensional cases and discuss the application of generalized Higuchi dimensions in these scenarios.
Paper Structure (7 sections, 8 theorems, 59 equations, 12 figures)

This paper contains 7 sections, 8 theorems, 59 equations, 12 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. The existence of the attractor by contractivity and approximation by partial attractors
  4. Fuzzy approach for idempotent invariant measures
  5. Higuchi fractal dimension via numerical approximations
  6. 1-D idempotent fractals and the Higuchi fractal dimension
  7. 2-D idempotent fractals and grey level images

Key Result

Theorem 2.2

A functional $\mu:C(X, \mathbb{R}) \to \mathbb{R}$ is an idempotent measure if and only if there exists $\lambda\in U(X, \mathbb{R}_{\max})$ satisfying There is a unique such function $\lambda$ in $U(X, \mathbb{R}_{\max})$ and $\mu\in I(x)$ if and only if $\oplus_{x\in X}\lambda(x) = 0.$

Figures (12)

  • Figure 1: Plot of the maps for the IFS $S_4$.
  • Figure 2: Approximation scheme to obtain $\nu$, where $\mu$ and $\eta$ are arbitrary initial idempotent measures.
  • Figure 3: Algorithm to compute the Higuchi fractal dimension for one dimensional series
  • Figure 4: Approximation scheme for $\mu_{5}$, $\mu_{10}$ and $\mu_{100}$.
  • Figure 5: HFD for $\hat{\lambda}_{5}(x)$, $\hat{\lambda}_{10}(x)$ and $\hat{\lambda}_{100}(x)$.
  • ...and 7 more figures

Theorems & Definitions (23)

  • Definition 2.1
  • Theorem 2.2
  • Definition 2.3
  • Definition 2.4
  • Example 2.5
  • Definition 2.6
  • Lemma 2.7
  • Definition 2.8
  • Definition 2.9
  • Definition 3.1
  • ...and 13 more