Kolakoski sequence: links between recurrence, symmetry and limit density

The Kolakoski sequence $S$ is the unique element of \(\left\lbrace 1,2 \right\rbrace^{\omega}\) starting with 1 and coinciding its own run length encoding. We use parity lengths particular subclasses initial words \(S\) as a unifying tool to address links between main open questions - recurrence, mirror/reversal invariance asymptotic density digits. In we prove that recurrence implies reversal invariance, give sufficient conditions which would imply 1s \(\frac{1}{2}\).