Imagine a system that evolves over time, for example the economy, the weather or the positions of the planets in the solar system. Some systems of this kind, for example the solar system, are deterministic in the sense that the state of the system now completely determines the state of the system in, say a year's time. One way of modelling such systems is to consider a function f mapping the space X of all possible states to itself.
In this approach, if the system is in the state x at time 0, then at time 1 it is in the state f(x), at time 2 it is in the state f(f(x))=f^2(x) and so on. Systems of this kind are called discrete dynamical systems. Typically, we assume that f is
a continuous function from X to itself and that X is a compact metric space, we refer to the states in X as points. In studying such systems one is often interested in how a particular point (or state) develops over time, so that one is interested in the orbit of the point, that is the sequence of points x, f(x), f^2(x), f^3(x), and so on. One might also be interested in the the limiting behaviour of a point x, i.e. the set of points to which the orbit converges, which is a measure of how the system behaves in the long term. In our example of the solar system, for example, given the current configuration of the solar system, we might be interested in whether in the future it gradually flies apart or collapse in to the sun or remains in some steady state.
Now suppose that we are trying to calculate the orbit of a point, for example using a computer. The computer will introduce rounding errors, so that each time we try to calculate f(x) we obtain a value that is close to, but not exactly equal to f(x). This leads us to the concept of a 'd-psuedo-orbits', that is a sequence of points x_1, x_2, x_3, and so on, with the property that the distance between f(x_n) (the actual value of x_n after one time step) and x_{n+1} (the computer calculated value) is less than d. It turns out that in some dynamical systems one can 'shadow' pseudo-orbits by real orbits, so that we know that the computed pseudo-orbit is not too far away from an actual orbit. Technically we say that f has the shadowing property provided that for any (potentially very small) e>0, there is a d>0 such that for every d-pseudo-orbit x_1, x_2, x_3, ... , there is a point z such that the distance from the nth point in the pseudo-orbit and the nt point in the orbit of z is no greater than e. This notion of shadowing has played an important role in the theory of dynamical systems. Furthermore, in an omega-limit set, for any d, there is a d-pseudo-orbit between any two points, which is to say that omega limit sets are 'internally chain transitive'.
The notion of shadowing is closely related to characterizations of omega-limit sets in terms of internal chain transitivity and the aim of the ShadOmIC project is to develop the general theory behind these relations. The project contributes to the state of the art in an important area of mathematics, the theory of dynamical systems. It brings a leading young mathematician from the US to Europe, establishing new international collaborations and enhancing the reputation and research capacity of the European Union.
