We begin our journey into smooth ergodic theory by constructing some
principal examples of dynamical systems (both with discrete and continu-
ous time), which illustrate various fundamental phenomena associated with
uniform as well as nonuniform hyperbolicity. We shall only consider smooth
systems which preserve a smooth measure (i.e., a measure which is equivalent
to the Riemannian volume) leaving aside an important class of dissipative
systems (e.g., the H´ enon attractor).
While experts in the field believe that hyperbolic behavior is typical
in some sense, it is usually diﬃcult to rigorously establish that a given
dynamical system is hyperbolic. In fact, known examples of uniformly hy-
perbolic diffeomorphisms include only those that act on tori and factors of
some nilpotent Lie groups and it is believed – due to the strong requirement
that every trajectory of such a system must be hyperbolic – that those are
the only manifolds carrying uniformly hyperbolic diffeomorphisms. Despite
such a shortage of examples (at least in the discrete-time case) the uniform
hyperbolicity theory provides great insight into many principal phenomena
associated with hyperbolic behavior and, in particular, to obtaining an es-
sentially complete description of stochastic behavior of such systems.
In quest for more examples of dynamical systems with hyperbolic be-
havior one examines those that are not uniformly hyperbolic but possess
“just enough” hyperbolicity to exhibit a high level of stochastic behavior.