Carleson Interpolation, Boundary Rigidity, and Classification of Spectral Transformations Preserving Diagonal Operator Orbit Frames

Maps $Φ:D\to D$ of the unit disk preserve frame generator sets for single orbits of normal diagonal operators exactly when they preserve Carleson interpolating sequences bidirectionally and satisfy $1-|Φ(z)|^2\asymp1-|z|^2$ uniformly. Denote this class by $\mathcal P$. Equality of frame generator se