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

TL;DR

This paper studies invariant measures for countable idempotent iterated function systems, introduces finite approximations, and explores their fractal dimensions, including numerical methods for higher dimensions.

Contribution

It introduces partially finite idempotent IFSs and proves convergence of their invariant measures to those of the original system, with applications to fractal dimension analysis.

Findings

Unique invariant idempotent probability with constant weights

Convergence of measures from finite to countable IFSs

Numerical approximation of Higuchi fractal dimensions in 2D

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.