The time evolution of this system is a flow, and if we discretize this flow we have a map of the phase space to itself. This setup occurs ubiquitously in models in science and mathematics. Most mathematical models of physical phenomena may be studied as dynamical systems. One of the attractions of working in dynamical systems is not only the wide applicability of the theory, from engineering and biological applications to number theory and other areas of pure mathematics, but also the wide variety of branches of mathematics that it uses. Modern dynamical systems theory uses techniques from analysis, differential geometry, probability theory, measure theory and group theory to name just a few.
Ergodic theory is the study of the statistical behaviour of a dynamical system.
It arose as a result of investigations in statistical mechanics and work by Poincare on the stability of the planetary orbits. Many physical systems are so disordered that a statistical description is perhaps the only one possible.