Axiom A

An Axiom A system is a dynamical system satisfying two conditions introduced by Stephen Smale in his 1967 paper on differentiable dynamical systems: the non-wandering set is hyperbolic, and the periodic points are dense in the non-wandering set. Axiom A systems admit a finite spectral decomposition into basic sets — attractors, repellers, and saddle-like sets — each of which is topologically transitive and can be encoded by a Markov partition. They were once believed to be generic among all dynamical systems, but the Newhouse phenomenon destroyed this hope, revealing that systems with infinitely many attractors are dense in certain regions. Axiom A remains the cleanest class of chaotic systems, but it is a cathedral in a wilderness of wilder dynamics.