In this lecture we prove the famous Poincaré Classification Theorem, which tells us that for transitive reversible dynamical systems on $S^1$, the rotation number is a complete dynamical invariant, in the sense that two such systems are conjugate if and only if they have the same rotation number.

Thus the rotation number is the “best possible” type of invariant for such systems. In comparison, topological entropy is the “worst possible” invariant (since it always vanishes for reversible systems on $S^1$, see Lecture 8).