Mathematical Conversations

In 1978, Charles Conley classified all continuous dynamical systems. His theorem, dubbed the "fundamental theorem of dynamical systems" states that the orbits of any continuous map on a compact metric space fall into two classes: gradient-like and recurrent. When the recurrent part is factored out, the dynamics appear to be gradient-like. While one might wonder how a theorem that applies to every continuous map could be of any use, it plays a foundational role in many deep results.