Chapter 1

Examples of

Hyperbolic

Dynamical Systems

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.

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