论文标题
具有超高关闭气缸II的迭代功能系统II
Iterated function systems with super-exponentially close cylinders II
论文作者
论文摘要
直到最近，在分形几何形状中，要确定是否存在作用于$ \ Mathbb {r} $的迭代功能系统，而没有确切的重叠，因此在所有小尺度上都超过了圆柱体。作者和Bárány和Käenmäki证明了满足这些属性的迭代功能系统。在本文中，我们证明了关于参数化家族中这种迭代函数系统存在的一般定理。该定理表明，如果一个参数化的家族包含两个独立的亚家族，并且导致精确重叠的一组参数满足了一些弱拓扑假设，则原始家族将包含满足所需属性的迭代功能系统。我们包括几个可以应用此定理的参数化家族的明确示例。
Until recently, it was an important open problem in Fractal Geometry to determine whether there exists an iterated function system acting on $\mathbb{R}$ with no exact overlaps for which cylinders are super-exponentially close at all small scales. Iterated function systems satisfying these properties were shown to exist by the author and by Bárány and Käenmäki. In this paper we prove a general theorem on the existence of such iterated function systems within a parameterised family. This theorem shows that if a parameterised family contains two independent subfamilies, and the set of parameters that cause exact overlaps satisfies some weak topological assumptions, then the original family will contain an iterated function system satisfying the desired properties. We include several explicit examples of parameterised families to which this theorem can be applied.
