 Kunen critical sequence

The critical sequence of $j$ starts with its critical point $\kappa$ and iterates ${\kappa }_{n+1}=j({\kappa }_n)$. Its supremum $\hat{\kappa }={\sup }_{n<\omega }{j}^n(\kappa )$ is the least fixed point of $j$ above $\kappa$ provided this sequence and its supremum belong to the domain and the embedding fixes $\omega$.