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.

翻译：我们研究了在紧致度量空间中定义的可数幂等迭代函数系统（IFS）的不变幂等概率集。我们证明，在权重恒定的情况下，存在唯一的不变幂等概率。利用Secelean处理可数IFS的方法，我们引入了部分有限幂等IFS，并证明了这些系统的不变幂等测度序列收敛于原始可数IFS的不变测度。随后，我们应用这些结果通过离散系统来逼近此类测度，并在一维情形下生成了可计算其Higuchi分形维数的数据序列。最后，我们给出了二维情形的数值逼近，并讨论了广义Higuchi维数在这些场景中的应用。