In dynamical systems, one considers the pair (X,T) where T is a map from a space X to itself.
We can view the map T as moving the points around X and apply it repeatedly, taking the point of view that the space X evolves over time.
There are several different subcategories of dynamical systems based on what kind of structure the set X has, and how much of it is preserved by T. We arrive at other important subcategories of dynamics by consideration of various (semi-) groups acting on X.
In addition to being an important subject in its own right, there are many examples of problems for which solutions became apparent only when the problem is rephrased in dynamical systems terms.
We focus on ergodic theory and symbolic dynamical systems (which model topological or smooth dynamical systems by a discrete space consisting of infinite sequences of abstract symbols and a shift operator).