RANDOM 2D LINEAR COCYCLES I: DICHOTOMIC BEHAVIOR

The paper’s main theorem states the exact dichotomy the model proves: for Mat^+_2-valued Bernoulli/primitive Markov cocycles with at least one singular and one invertible component, either the cocycle is projectively uniformly hyperbolic (PUH) or it can be approximated by cocycles with L1 = −∞ (Theorem 1.1) . The paper establishes PUH ⇔ existence of an invariant multi-cone (Theorem 2.1) and develops a rank‑1 criterion via the sets W+ and W− (Theorem 3.2) that connects null words with loss of PUH and with L1 = −∞ (see also Theorem 4.2 and Example 5.2) . The candidate solution gives a constructive alignment/perturbation scheme: non‑PUH implies arbitrarily close lines from forward/backward branches, then a small adjustment to a single invertible matrix makes a mortal (null) word, forcing L1 = −∞. This is consistent with the paper’s W+/W− mechanism and with the density/residuality of L1 = −∞ outside PUH in analytic families (Corollary 4.3), but it is a different presentation emphasizing local projective alignment and a one-letter perturbation rather than the paper’s analytic-family and multi-cone framework . Hence both are correct, with materially different proof styles.