Inverse limits of unimodal maps and sphere homeomorphisms

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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.