Structural Stability in Piecewise Möbius Transformations

The paper’s Theorem 11 asserts that if all component maps are loxodromic, F is hyperbolic and α‑expanding, and B(F) moves holomorphically (B‑structural stability), then F is structurally stable. The published proof builds a holomorphic motion on a canonical subset E and extends it to a quasiconformal conjugacy via the Bers–Royden extension, verifying conjugacy on all parts of the phase space. This argument is coherent with the paper’s definitions (including that “hyperbolic” already excludes wandering regular components) and is internally consistent, though it seemingly does not use the α‑expanding hypothesis in an essential way, making the sufficiency statement stronger than necessary. The candidate solution provides an alternate construction: pull back the boundary motion to the entire pre‑discontinuity set, extend to α(F) by contraction using α‑expansion, linearize near attracting cycles (Hartman–Grobman), and glue. Under the paper’s hyperbolicity (which includes no wandering components) and α‑expansion, this approach is also correct. Net: both are valid, with substantially different techniques.