Translation numbers for bundle automorphisms and a cocycle on a group of homeomorphisms

The paper’s Theorem 1.1 shows that if X is compact and α is a standard representative, then any bundle automorphism ĝ with a nonzero local or mean translation number is undistorted, via a seminorm ‖ĝ‖α = supx|ρx,α(ĝ)|, subadditivity, and the bound C|ĝ|S ≥ ‖ĝ‖α, yielding τ(ĝ) ≥ |rotx,α(ĝ)|/C > 0; mean ≠ 0 implies some x with local ≠ 0 by Remark 2.7(1) . The candidate solution defines the same displacement function ρx,α (as Φ), proves the same cocycle identity and seminorm inequality, and derives linear growth for ‖ĝn‖ from either a nonzero local translation number or, equivalently, by integrating the telescoping identity against a g-invariant measure (an equivalent route to the paper’s reduction). Thus, both are correct and essentially the same proof strategy, differing only in how the mean case is handled .