In short terms, ergodic theory is the mathematical discipline that deals with dynamical systems endowed with invariant measures. Let us begin by explaining what we mean by this and why these mathematical objects are so worth studying. Next, we highlight some of the major achievements in this field, whose roots go back to the physics of the late 19th century. Near the end of the preface, we outline the content of this book, its structure and its prerequisites.

What is a dynamical system?

There are several definitions of what a dynamical system is some more general than others. We restrict ourselves to two main models.

The first one, to which we refer most of the time, is a transformation f : M → M in some space M. Heuristically, we think of M as the space of all possible states of a given system. Then f is the evolution law, associating with each state x ∈ M the one state f(x) ∈ M the system will be in a unit of time later. Thus, time is a discrete parameter in this model.

We also consider models of dynamical systems with continuous time, namely flows. Recall that a flow in a space M is a family f t : M→M, t ∈ ℝ of transformations satisfying

f0 = identity and f t ∘ f s = f t+s for all t, s ∈ ℝ.

Flows appear, most notably, in connection with differential equations: take f t to be the transformation associating with each x ∈ M the value at time t of the solution of the equation that passes through x at time zero.

We always assume that the dynamical system is measurable, that is, that the space M carries a σ-algebra of measurable subsets that is preserved by the dynamics, in the sense that the pre-image of any measurable subset is still a measurable subset. Often, we take M to be a topological space, or even a metric space, endowed with the Borel σ-algebra, that is, the smallest σ-algebra that contains all open sets. Even more, in many of the situations we consider in this book, M is a smooth manifold and the dynamical system is taken to be differentiable.