A wide range of problems in complex dynamics reduce to showing lower bounds on certain conformal moduli. These include: local connectivity of Julia sets, rigidity and local connectivity in parameter space, a priori bounds for renormalization, and precompactness of hyperbolic components. Many of these have been solved using the near degenerate regime, where we describe in detail what may happen when these moduli are close to zero. I will try in the hour to outline the history, principles, and applications of the near degenerate regime, and also describe its ultimate realization as the “totally degenerate regime”.
