Introduction to Hyperbolic Sets

One of the most illuminating general observations about dynamical systems is that it often happens that orbits starting very close together diverge exponentially. Exponential separation of orbits gives rise, notably, to the sensitive dependence on initial condition which accounts for the apparently stochastic behavior of deterministic dynamical systems. In these lectures, I will discuss systems which have a technically very strong version of the property of exponential separation of orbits, ones in which there are no neutral separations between nearby orbits so that each pair separates exponentially either forward or backward in time (and most separate both forward and backward) with strong uniformity assumptions on the rate of separation. Separation of orbits will not, however, be required everywhere in the state space of the system, but only in the neighborhood of some compact invariant set. Thus, the objects we will analyze are, roughly, compact invariant sets with the property that pairs of orbits starting out very near to each other and remaining near the set in question diverge exponentially in a uniform way. Such sets are called hyperbolic sets. (The above is intended only as a very general indication of what a hyperbolic set is and will be misleading if taken too literally; the formal definition is given in Section 3.)