Inverse limits of unimodal maps and sphere homeomorphisms

离散动力系统是按一定规律按时间步长演化的系统。也就是说，如果我们知道系统今天处于什么状态，那么就有一个规则来决定它明天将处于什么状态（以及后天、后天……）。这项研究从称为单峰映射的非常简单的模型系统开始，其中系统的“状态”由单个数字指定-通常不是整数-并且规则的形式特别简单。在20世纪70年代，这样的系统推动了现代动力系统理论中最早的一些步骤。虽然我们可以通过重复应用这一规则使离散动力系统向前运行，但我们通常不能向后运行，因为昨天可能有两种不同的状态，而今天可能会导致相同的状态。如果我们今天处于那个状态，我们不能仅仅通过研究系统就知道我们昨天处于什么状态：我们必须记住我们在哪里。所以如果我们想让系统像向前运行一样容易地向后运行，我们需要保存系统过去的每一个状态的列表，也就是说，保存关于过去状态和现在状态的信息。所有这些“有历史的状态”的集合被称为动力系统的“逆极限”。我们为能够在时间上倒退而付出的代价是，关于系统状态的信息变得更加复杂：无限多的数字，而不仅仅是一个，它们组合在一起，形成了具有极大拓扑复杂性的空间，比如所谓的不可分解连续体。在过去的二三十年里，人们对这些空间进行了大量的研究。最近我们有了一个意想不到的发现，这些拓扑复杂的空间可以通过一些轻微的手术来产生非常简单的空间：二维球体。有了这一见解，时间可逆单峰映射被认为与另一类系统密切相关，这类系统在动力系统理论中有单独的兴趣：球同胚。动力系统的丰富性通常只有在我们研究依赖于一个参数的逐渐变化的系统族，并观察随着参数的逐渐变化，其行为是如何变化的（有时是非常突然的）时，才能得到适当的理解。我们最近发现的一个主要缺陷是，我们没能证明，当我们的单峰映射逐渐变化时，与之相关的球同胚也会逐渐变化。这项研究的目的就是填补这一空白。

A discrete dynamical system is a system which evolves in time steps according to a rule. That is, if we know what state the system is in today, there is a rule to decide what state it will be in tomorrow (and hence the day after, and the day after that...). This research starts with very simple model systems called unimodal maps, where the "state" of the system is specified by a single number - not normally a whole number - and the rule is of a particularly simple form. Such systems were a driving force for some of the earliest steps in the modern theory of dynamical systems during the 1970s.While we can run a discrete dynamical system forwards in time by repeatedly applying the rule, we can't normally run it backwards, since there may be two different states yesterday which result in the same state today. If we're in that state today, we can't know what state we were in yesterday just by studying the system: we have to remember where we were. So if we want to run the system backwards in time as easily as we run it forwards, we need to keep a list of every state that it's been in in the past: that is, to hold information about past states as well as present state. The collection of all such "states with a history" is called the "inverse limit" of the dynamical system.The price we pay for being able to go backwards in time is that the information about the state of the system has become much more complicated: infinitely many numbers, instead of just one, which fit together to give spaces of great topological complexity, such as so-called indecomposable continua. These spaces have been much studied over the last 20 or 30 years.Recently we made the unexpected discovery that these topologically complicated spaces can be subjected to some mild surgery to yield very simple spaces: two-dimensional spheres. With this insight, time-reversible unimodal maps are seen to be very closely related to another class of systems which have been of separate interest in dynamical systems theory: sphere homeomorphisms.The richness of dynamical systems can often only be properly appreciated when we study gradually varying families of systems, dependent on a parameter, and see how the behaviour changes (sometimes quite suddenly) as the parameter changes gradually. A major gap in our recent discovery is that we haven't been able to show that, when our unimodal map changes gradually, so does the sphere homeomorphism associated with it. The aim of this research is to plug that gap.