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We discuss in this lecture, what dynamical systems are and where the subject is located within mathematics.
The theory of dynamical systems deals with the evolution of systems. It describes processes in motion, tries to predict the future of these systems or processes and understand the limitations of these predictions.
All areas of mathematics are linked together in some way or an other. Intersections of fields like algebraic topology, geometric measure theory, geometry of numbers or algebraic number theory can be considered full blown independent subjects. The theory of dynamical systems has relations with all other main fields and intersections typically form subfields of both.
Link with algebra: group theorists often look at the action of the group on itself. The action of the group on vector spaces defines a field called representation theory.
Link with measure theory: in ergodic theory one studies a map T on a measure space (X, µ). Measure theory is one foundation of ergodic theory.
Link with analysis: the study of partial differential equations or functional analysis as well as complex analysis or potential theory.
Link with topology: the Poincare conjecture states that every compact three dimensional simply connected manifold is a sphere. The problem is currently attacked using a dynamical system on the space of all surfaces which is called the Ricci flow.
Link with geometry: Kleins Erlanger program attempted to classify geometries by its symmetry groups. For example, the group of projective transformations on a projective space.
A concrete dynamical system in geometry is the geodesic flow. An other connection
is the relations of partial differential equations with intrinsic geometric properties of the
space.
Link with probabilit y theory: sequences of independent random variables can be obtained using dynamical systems.
For example, with T(x) = 2x mod 1 and with the function f which is equal to 1 on [0, 1/2] and equal to 0 on [1/2, 1], f(T n(x)) are independent random variables for most x.
Link with logic: logical deductions in a proof or doing computations can be modeled as dynamical systems. Because every computation by a Turing machine can be realized as a dynamical system, there are fundamental limitations, what a dynamical system can compute and what not.
Link with number theory: some problems in the theory of Diophantine approximations can be seen as problems in dynamics.
For example, if you take a curve in the plane and look at the sequence of distances to nearest lattice points, this defines a dynamical system.
A final link: a category X of mathematical objects has a semigroup G of homomorphisms acting on it (topological spaces have continuous maps, sets have arbitrary maps, groups, rings fields or algebras have homomorphisms, measure spaces have measurable maps).
We can view each of these categories as a dynamical system. One can even include the category of dynamical systems with suitable homomorphisms. But this viewpoint is not a very useful in itself