论文标题
迭代功能系统吸引子的过渡现象
Transition Phenomena for the Attractor of an Iterated Function System
论文作者
论文摘要
迭代功能系统（IFSS）及其吸引子几乎从其成立开始就成为分形几何学理论的核心。 IFS中功能的合同性是迭代函数系统理论的核心。如果IFS中的功能是收缩，则保证IFS具有独特的吸引子。但是，最近，在合同缺陷和IFS扩展之间的吸引子发生的吸引子发生的情况上引起了人们的兴趣。那是本文的主题。对于一个家庭，根据真实参数$ t> 0 $，研究了两种类型的过渡吸引子的存在和属性，称为下过渡吸引子$ a _ {\ bullet} $和上过渡吸引子$ a^{\ buttral} $。一个主要定理指出，对于众多的IFS系列，有一个阈值$ t_0 $，因此IFS $ f_t $具有唯一的吸引子$ a_t $ for $ t <t_0 $，而没有$ t> t_0 $的吸引子。在阈值$ t_0 $上，有$ f_ {t_0} $ - 不变设置$ a^{\ bullet} $，这样$ a^{\ bullet} = \ lim_ {t \ rightarrow t_0} a_t $。
Iterated function systems (IFSs) and their attractors have been central to the theory of fractal geometry almost from its inception. And contractivity of the functions in the IFS has been central to the theory of iterated functions systems. If the functions in the IFS are contractions, then the IFS is guaranteed to have a unique attractor. Recently, however, there has been an interest in what occurs to the attractor at the boundary between contractvity and expansion of the IFS. That is the subject of this paper. For a family $F_t$ of IFSs depending on a real parameter $t>0$, the existence and properties of two types of transition attractors, called the lower transition attractor $A_{\bullet}$ and the upper transition attractor $A^{\bullet}$, are investigated. A main theorem states that, for a wide class of IFS families, there is a threshold $t_0$ such that the IFS $F_t$ has a unique attractor $A_t$ for $t<t_0$ and no attractor for $t>t_0$. At the threshold $t_0$, there is an $F_{t_0}$-invariant set $A^{\bullet}$ such that $A^{\bullet} = \lim_{t\rightarrow t_0} A_t$.
