Conley's fundamental theorem of dynamical systems

Remarks

The proof uses attractor–repeller pairs $(A, A^\ast)$. For each such pair, Conley builds a continuous function equal to $0$ on $A$ and to $1$ on $A^\ast$ that strictly decreases along orbits between them. A compact metric space carries only countably many attractors $A_1, A_2, \ldots$, and $V = \sum_n 3^{-n} V_n$ combines the corresponding functions. The key fact is that $\mathcal{R}(\varphi)$ is the intersection of the sets $A \cup A^\ast$ over all attractor–repeller pairs. Example: let $f$ be a Morse function on a compact Riemannian manifold with pairwise distinct critical values. For the flow of $-\nabla f$, the chain recurrent set is the set of critical points, and $V = f$ satisfies the three conditions. Compactness is needed for the countability of attractors and for the existence of the attractor–repeller decomposition used in the proof.

Going deeper

For a student

The function $V$ orders the chain components like levels of a landscape and shows that orbits outside $\mathcal{R}(\varphi)$ go from higher components to lower ones. This is why the result is sometimes called the fundamental theorem of dynamical systems: the chain components are the indecomposable pieces, and between them there is only gradient-like motion. The statement is the starting point for Morse decompositions and for the Conley index. Versions for continuous maps of compact metric spaces, with $\varepsilon$-chains in place of $(\varepsilon, T)$-chains, also hold.