COVER TIMES IN DYNAMICAL SYSTEMS

The paper proves sharp two-sided bounds for the expected cover time E_mu(τ_δ) in terms of M_μ(δ) using a uniform spectral-perturbation approach for transfer operators with holes, careful control of short returns, and a Matthews-style reduction from covering to maximal hitting times; see Theorems 2.1–2.2 and the surrounding machinery in Sections 2–5 . The candidate solution sketches a different route via Abadi-type ψ-mixing hitting-time bounds for arbitrary unions of n-cylinders, a cylinder/ball sandwich, and a union-bound over a δ-grid. However, it assumes uniform exponential hitting-time estimates for all unions of n-cylinders with constants depending only on μ(A), without addressing the delicate uniform short-return phenomena that the paper treats explicitly (buffers and symbolic analysis). Under the paper’s assumptions, those uniform bounds are not automatic from ψ-mixing alone. Because these missing uniformity and short-return controls are essential to make the grid/union-bound argument rigorous in this generality, the model’s proof is incomplete/incorrect as stated, while the paper’s proof is correct.