Spectral theoretic characterisation of Markov chain convergence

The paper proves L1 convergence of densities to the absolutely continuous invariant measures for the logistic map (r=4) and the generalized logistic/Chebyshev maps, with exponential rates O(2^{-n}) and O(m^{-n}) for BV initial data (Theorems 3.1 and 3.2), via a spectral–theoretic construction and a piecewise-linear folding map whose Frobenius–Perron action contracts variation; see the theorem statements and the BV estimates in Section 3.3 . The candidate solution proves the same results by an explicit semiconjugacy to angle-multiplication and a direct transfer-operator argument, recovering the same invariant densities and rates. Minor fix: in the step-function argument one should approximate by m-adic step functions so that K | m^n holds eventually. The core results and quantitative rates coincide; the proofs are different in emphasis (spectral vs. classical PF/semiconjugacy). Conjugacy to tent/Chebyshev maps in the paper corroborates the candidate’s trigonometric semiconjugacies and invariance of the arcsine density , and the paper’s “Frobenius–Perron operator” subsection aligns with the model’s operator-based reasoning .