RANDOM AND MEAN LYAPUNOV EXPONENTS FOR GLn(R)

The counterexample uses a measure µ supported on orthogonal conjugates of a fixed shear, i.e., µ is O(n)-conjugation-invariant. The paper’s Theorem 1 is proved for “orthogonally invariant” measures in the sense used in their argument: invariance under the left O(n)-action (indeed, they disintegrate along left O(n)-orbits and apply an inequality that integrates over U with A replaced by UA). This orbit-by-orbit step, and the fact that |det A|g| is constant on each left O(n)-orbit, are explicit in Theorem 2 and the proof of Theorem 3, and they do not hold for conjugation orbits. Hence the candidate’s measure does not satisfy the paper’s hypotheses, so no contradiction is obtained. See Theorem 1 and its reduction via Theorems 2–3 and the orbit argument .