Continuous itinerary functions and dendrite maps
Continuous itinerary functions and dendrite maps
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DOI：
10.1016/j.topol.2007.04.001
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发表时间：
2007-08
影响因子：
0.6
通讯作者：
S. Baldwin
中科院分区：
文献类型：
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作者：
S. Baldwin
我们研究了树突上映射的“捏合序列”理论，集中于那些具有一个“转折点”和“唯一行程性质”的映射（即，不同的点有不同的路线）。这个理论与多项式Julia集和Hubbard树的理论有很大的重叠，但也与这两个理论有很大的不同（唯一的行程性质并不总是成立）。我们发现，独特的行程属性是一个强大的属性，并允许一个简单的分类，这样的树突地图相对于他们的揉捏序列（共轭，如果没有非平凡不变的subdendritic）。这里介绍的主要工具之一是连续行程功能。如果一个人把用来定义行程的所有符号序列的集合相对于一个空间的划分，有一个自然的拓扑，它迫使行程函数（从原始空间到符号序列的空间）是连续的，尽管这往往导致非豪斯多夫行程拓扑。尽管有这个明显的缺点，我们表明，这种行程拓扑是一个有用的工具，用于分析度量空间上的连续映射的动态。给出了这些映射的拓扑性质的捏合序列的刻画，包括各种传递性性质，逆极限的可分解性，以及这些映射的某些“分段线性化”的存在性，这些“分段线性化”是区间上“帐篷”映射的自然推广.行程拓扑为所有捏合序列的参数空间提供了一个自然拓扑，其中的一个自然子空间将被证明具有一个单点紧化，这是一个树突，与Mandelbrot集有一个有趣的连接。
We examine the “kneading sequence” theory of maps on dendrites, concentrating on those maps having one “turning point” and the “unique itinerary property” (i.e., distinct points have distinct itineraries). This theory has major overlaps with the theories of polynomial Julia Sets and Hubbard Trees, but also has significant differences from those two theories (for which the unique itinerary property does not always hold). We show that the unique itinerary property is a powerful property, and allows a simple classification of such dendrite maps with respect to their kneading sequences (up to conjugacy if there is no nontrivial invariant subdendrite). One of the major tools introduced here is the continuous itinerary function. If one takes the set of all sequences of the symbols used to define the itineraries with respect to a partition of a space, there is a natural topology which forces the itinerary function (from the original space into the space of sequences of symbols) to be continuous, although this often leads to a non-Hausdorff itinerary topology. Despite this apparent drawback, we show that this itinerary topology is a useful tool for analyzing the dynamics of continuous maps on metric spaces. Characterizations in terms of kneading sequences are given for topological properties of these maps, including various transitivity properties, (in)decomposability of inverse limits, and the existence of certain “piecewise linearizations” of such maps which are a natural generalization of “tent” maps on the interval. The itinerary topology provides a natural topology for the parameter space of all kneading sequences, a natural subspace of which will be shown to have a one-point compactification that is a dendrite, with an interesting connection to the Mandelbrot Set.