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. 3 https://doi.org/10.1090//gsm/148/01

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