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A theorem of Guglielmi and Zennaro implies that if the uniform norm growth of a locally constant-cocycle on the full shift is not exponential, then it must be either bounded or linear, with no other possibilities occurring. We give an alternative proof of this result and demonstrate that its conclusions do not hold for Lipschitz continuous cocycles over the full shift on two symbols.

A note on the marginal instability rates of two-dimensional linear cocycles