Lasota and Myjak demonstrated that one can study attractors to an iterated function system (IFS) possibly containing discontinuous functions. We continue this line of thought by working with IFS with a lower semicontinuous Hutchinson-Barnsley operator and two new (but not significantly different) types of attractors, the small attractor (the smallest closed nonempty invariant set) and minimal attractors (a minimal closed nonempty invariant set). We characterize exactly when an IFS possesses a small attractor and provide several practically verifiable sufficient conditions for this. To study minimal attractors we create the notion of the weak basin; we show a minimal attractor behaves much like a small attractor on its weak basin and that the weak basin is the largest set in which a minimal attractor behaves like this. We then give a characterization of when the weak basin is open. Further, we show that, when the iterates of the Hutchinson-Barnsley operator of an IFS form an equicontinuous set of multifunctions, both the weak basin and the point wise basin (of a point wise attractor) is closed.

Lasota和Myjak证明了人们可以研究可能包含不连续功能的迭代功能系统（IFS）的吸引子。我们通过与IFS一起使用较低的半连续Hutchinson-Barnsley算子和两种新的（但差异不大）吸引子，小吸引子（最小封闭非空不变集）和最小吸引子（最小封闭非空吸引子）来继续这种思路。不变集）。当IFS拥有小的吸引子时，我们可以准确地描述其特征，并为此提供几个可实际验证的充分条件。为了研究最少的吸引子，我们创建了弱盆地的概念。我们显示出最小吸引子的行为与弱盆地上的小吸引子的行为非常相似，并且弱盆地是最小吸引子表现为此类行为的最大集合。然后，我们给出弱化盆地何时开放的特征。进一步，我们表明，当IFS的Hutchinson-Barnsley算子的迭代形成一个等连续的多功能集时，弱盆和点状吸引子的点状盆都将关闭。