We construct a sequence of finite sets which approximates, with arbitrarily small and controlled error, the attractor set of a (finite or countable) given iterated function system of order m. Our main tool are the so-called \(\alpha \)-dense curves which are from the Hausdorff–Pompeiu distance point of view, a generalization of the space-filling curves. Also we prove that, under suitable conditions, our result can be used to approximate infinite-dimensional fractals generated by integral equations of Volterra type.

中文翻译：

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用$$ \ alpha $$α-密集曲线逼近m阶迭代函数系统的吸引子集

我们构造了一个有限集序列，该序列以给定的 m阶迭代函数系统近似地（有限或可数）的吸引子集具有任意小的受控误差。我们的主要工具是所谓的\（\ alpha \）-密集曲线，从Hausdorff–Pompeiu距离的角度出发，是对空间填充曲线的概括。我们还证明，在合适的条件下，我们的结果可用于逼近由Volterra型积分方程生成的无限维分形。