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INTRODUCTION TO THE HABILITATIONSSCHRIFT: ALGEBRAIC METHODS IN HIGHER-DIMENSIONAL DYNAMICS

The theory of dynamical systems has developed in the last century from the beginning to a rich and broad field of mathematics. The main idea of dynamical systems is the idea of a time-evolving system. Although time-evolving systems have been studied from the very beginning of mathematical physics, the mathematical idea is to study them not in particular examples, but in a general theory. This has been motivated, for instance, by the non-predictability of examples like the 3-body problem: Assume we have three bodies in space, say a solar system with just the sun, the earth and the moon, and we want to find a formula telling us what will happen in time under the force of gravity. Even though this question leads to a well known differential equation, a general solution is not known. However, this is not the end of the problems in that example. Even if we had the possibility to simulate the behaviour of the system with the mass, position and velocity given (say with a computer of infinite accuracy), small errors in the initial data would make a huge difference after a surprisingly small time. This effect is known as chaos and has lead to many different mathematical notions of this behaviour (hyperbolicity, expansiveness). Since this concrete example cannot be simulated in its long term behaviour to answer questions like ‘Will the moon fall onto the earth?’, the question ‘What can be said about a system in general?’ gains interest. To start the mathematics we need to specify the meaning of the term dynamical system. We fix a space X, together with some mathematical structure (like a topology, group-structure or a probability measure). Since we are mainly interested in the long term behaviour of the system, we work with discrete time (instead of continuous time as in dynamical systems arising from differential equations). This means that we work with a map T : X → X which describes the position Tx of the point x ∈ X after one unit of time. Depending on the structure of the set X, we will have to require some properties of T . Furthermore it is often convenient and not much of a restriction to assume that T is invertible. Since a theory lives with (and dies with a lack of) examples, we now present some simple but illustrative examples of dynamical systems (X,T ). We start with topological dynamical systems where X will always be a compact metric space and T will be a continuous map. Then we will consider measurable dynamical systems, where X is a probability space and T is