Discontinuity of Lyapunov Exponents Near Fiber Bunched Cocycles

We give examples of locally constant $SL(2,\mathbb{R})$-cocycles over a Bernoulli shift which are discontinuity points for Lyapunov exponents in the H\"older topology and are arbitrarily close to satisfying the fiber bunching inequality. Backes, Brown, and the author have shown that the Lyapunov exponents vary continuously when restricted to the space of fiber bunched H\"older continuous cocycles. Our examples give evidence that this theorem is optimal within certain families of H\"older cocycles.