论文标题
关于图形吸引子的自相似性的二分法
A dichotomy on the self-similarity of graph-directed attractors
论文作者
论文摘要
本文寻求条件，以确保导向迭代功能系统（GD-ifs）的吸引子无法实现为标准迭代功能系统（IFS）的吸引子。对于一个牢固连接的有向图，众所周知，如果所有有向电路都通过顶点，那么基于图形的$ \ mathbb {r} $上的任何相似之处的GD-if-if-ifs相似之处，并满足凸开放式环境（COSC）（COSC），其吸引子与此顶点相关联，它也是（COSC）的吸引者（COSC）标准的吸引者。在本文中，我们显示以下互补结果。如果有向电路没有通过顶点，则存在基于图的GD-ifs，因此与该顶点相关的吸引子不是任何相似标准IF的吸引子。的确，我们为此类GD-IF吸引子提供代数条件，而不是标准IFS的吸引子，因此表明基于图的“几乎所有” cosc gd-ifss具有与该顶点相关的吸引子，这些吸引子与任何COSC标准IF的吸引者相关。
This paper seeks conditions that ensure that the attractor of a graph directed iterated function system (GD-IFS) cannot be realised as the attractor of a standard iterated function system (IFS). For a strongly connected directed graph, it is known that, if all directed circuits go through a vertex, then for any GD-IFS of similarities on $\mathbb{R}$ based on the graph and satisfying the convex open set condition (COSC), its attractor associated with this vertex is also the attractor of a (COSC) standard IFS. In this paper we show the following complementary result. If a directed circuit does not go through a vertex, then there exists a GD-IFS based on the graph such that the attractor associated with this vertex is not the attractor of any standard IFS of similarities. Indeed, we give algebraic conditions for such GD-IFS attractors not to be attractors of standard IFSs, and thus show that `almost-all' COSC GD-IFSs based on the graph have attractors associated with this vertex that are not the attractors of any COSC standard IFS.
