Hyperbolic Dynamics

Hyperbolic dynamics is the branch of dynamical systems theory that studies systems in which phase space can be decomposed, at every point, into expanding and contracting directions. A hyperbolic system is one in which trajectories that start close together diverge exponentially in some directions and converge exponentially in others, with no neutral directions — no directions in which trajectories neither expand nor contract. This property, introduced and developed by Stephen Smale in the 1960s, provides the rigorous framework within which chaos can be analyzed and classified.

The significance of hyperbolicity is that it makes chaotic systems predictable in a statistical sense even when individual trajectories are unpredictable. Hyperbolic systems possess Markov partitions that allow their dynamics to be encoded as symbolic dynamics, transforming continuous chaos into combinatorial structure. The Anosov diffeomorphisms — globally hyperbolic systems on manifolds — and the Axiom A systems introduced by Smale remain the best-understood classes of chaotic dynamical systems. Hyperbolic dynamics connects to ergodic theory through the study of invariant measures and to structural stability through the proof that hyperbolic systems form open sets in the space of all dynamical systems.