Mathematical consequences of eigenvalue rings in phase-dependent relaxation

Determine the mathematical consequences of rings of badly conditioned eigenvalues in truncated CMV matrices and explain why their effects differ between phases I and II.

Background

For CMV matrices with exponentially decaying Verblunsky coefficients, the eigenvalues of the finite truncated propagator can condense on a ring of radius μ. In phase II, this ring is composed of badly conditioned eigenvalues and obscures the true Ruelle–Pollicott resonance, whose modulus is smaller than μ. The authors observe that the same ring has different dynamical consequences in phases I and II, but do not provide a mathematical explanation of this distinction.