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 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. 3
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