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

研究的目的是寻找一种合适的方法，用符号动力学来表示非线性系统的动力学。对于二维及二维以上的微分同态，构造行程表示是相当困难的，这主要是因为对于微分同态，不存在一维地图的临界点等特殊点。Henon映射是最简单的非线性微分同构，对于Henon映射有一个“剪枝前”的思想，即可以将非马蹄Henon映射的非游走集看作是双符号全移的子移。1990年，Davis-MacKay-Sannami提出了一种机制，并给出了“缺失块表达式”（missing blocks expression）在非漫游集上的动态机制及其有希望的证据。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。但是，所有这些例子都是复杂的全马蹄形情况，并且对于具有切线和汇的情况，缺失块表达式（以及修剪前表示）尚不清楚。在这项研究中，我调查了有水槽的情况，我成功地找到了这种情况下缺失的块表达式。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，某个缺失的块只给出了一个周期为3的周期轨道，这在Henon地图上是不可能出现的。真实Henon地图的缺失块表达一定有更多的限制。沿着本研究的方向，我们可以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.