Metrics on Trees I. The Tower Algorithm for Interval Maps.

The paper’s Theorem 4.1(1) asserts that for any ϕ ∈ BV the unnormalized sequence L^nϕ converges to an eigenvector of eigenvalue λ. This is incorrect for λ > 1, since ‖L^nϕ‖ typically grows like λ^n. The proof the paper gives actually uses normalization by Mn = ‖(f?)^n(m0)‖ and a spectral projector P (so L^nχJ/Mn → c·PχJ), which supports the correct normalized statement λ^{-n}L^nϕ → Pϕ. The model explicitly makes this normalization and otherwise reproduces the convergence of Hn, the semiconjugacy to a constant-slope map, and the uniform convergence fn → f∞ correctly. See Theorem 4.1 and its proof for the paper’s statement and use of Mn, and the spectral decomposition and high-entropy unimodal gap that underpin the argument (duality and setup; spectral decomposition; unimodal high-entropy spectral gap; iteration via mn and Hn; semiconjugacy via linearly expanded metrics; conditional convergence of gn, hn) .