Ergodic theory studies the behavior of dynamical systems with respect to measures that remain invariant under time evolution. Indeed, it aims to describe those properties that are valid for the trajectories of almost all initial states of the system, that is, all but a subset that has zero weight for the invariant measure. Our first task, in Section 1.1, will be to explain what we mean by ‘dynamical system’ and ‘invariant measure’.

The roots of the theory date back to the first half of the 19th century. By 1838, the French mathematician Joseph Liouville observed that every energy-preserving system in classical (Newtonian) mechanics admits a natural invariant volume measure in the space of configurations. Just a bit later, in 1845, the great German mathematician Carl Friedrich Gauss pointed out that the transformation

(0,1]→ ℝ, x ↦ fractional part of 1/x,

which has an important role in number theory, admits an invariant measure equivalent to the Lebesgue measure (in the sense that the two have the same zero measure sets). These are two of the examples of applications of ergodic theory that we discuss in Section 1.3. Many others are introduced throughout this book.

The first important result was found by the great French mathematician Henri Poincaré by the end of the 19th century. Poincaré was particularly interested in the motion of celestial bodies, such as planets and comets, which is described by certain differential equations originating from Newton's law of universal gravitation. Starting from Liouville's observation, Poincaré realized that for almost every initial state of the system, that is, almost every value of the initial position and velocity, the solution of the differential equation comes back arbitrarily close to that initial state, unless it goes to infinity. Even more, this recurrence property is not specific to (celestial) mechanics: it is shared by any dynamical system that admits a finite invariant measure. That is the theme of Section 1.2.

The same theme reappears in Section 1.5, in a more elaborate context: there, we deal with any finite number of dynamical systems commuting with each other, and we seek simultaneous returns of the orbits of all those systems to the neighborhood of the initial state. This kind of result has important applications in combinatorics and number theory, as we will see.