To what extent can symbolic dynamics represent the structure of non-linear dynamics?

研究的目的是寻找一种用符号动力学表示非线性系统动力学的合适方法。对于二维及二维以上的单态映射，构造一个行程表示是相当困难的，主要是因为单态映射不存在像一维映射的临界点那样的特殊点。对于Henon映射这一最简单的非线性同态，有一种称为“剪枝前沿”的思想，即把非马蹄形Henon映射的非游荡集看作是双符号全移位的子移位. Davis-MacKay-Sannami在1990年提出了一种机制，证明了非游荡集上的动力学可以用“缺失块表达式”来表示，这是一种修剪前沿。最近，Arai给出了一个严格的证明，包括许多其他参数值的情况下。但是，所有这些例子都是复杂的全马蹄铁的情况，并且对于具有切线和汇的情况，缺失块表达式（以及因此修剪前表示）还不清楚。在这项研究中，我调查的情况下，有汇，我成功地找到了一个缺失块表达式这种情况下。目前，我的方法给出了几乎任何有限轨道的缺失块表达式。例如，某个缺失块只给出一个周期为3的周期轨道，这对于Henon映射是不可能发生的。真实的Henon映射的缺块表示必然有更多的限制，沿着本研究所得到的方向，我们有可能在非线性系统动力学的符号表示这一重要问题上取得一些进展。

The purpose of the research is to find an appropriate method to represent the dynamics of non-linear systems by using symbolic dynamics.For diffeomophisms of dimension two and more, it is quite difficult to construct an itinerary representation, mainly because for diffeomorphisms there exists no special point such as critical point of one-dimensional maps. For the Henon map, which is the simplest non-linear diffeomorphism, there is an idea called “pruning front" which is that one may regard the non-wandering set of non-horseshoe Henon map as a subshift of two-symbols full-shift. In 1990, Davis-MacKay-Sannami gave a mechanism and its promising evidence for the dynamics on the non-wandering set being represented by “missing blocks expression", which is a kind of pruning front. Recently, Arai gave a rigorous proof for that including many other parameter values cases. But, those all examples are the case of complex full-horseshoe, and missing blocks expression (and so pruning front representation) for the cases of having tangencies and sinks have not been known. In this research, I investigated the case of having sinks, and I succeeded to find a missing blocks expression for such case. For the moment, my method gives missing blocks expression for almost any finite orbit. For example, a certain missing blocks gives only one periodic orbit of period 3, which can not occur for the Henon map. There must be more restrictions for missing blocks expression of the real Henon map.By pursuing the direction obtained by this research, we may make some progress in the important problem of giving a symbolic representation to dynamics of non-linear systems.