Discontinuity of Lyapunov Exponents Near Fiber Bunched Cocycles

Abstract

We give examples of locally constant $SL(2,\mathbb{R})$-cocycles over a Bernoulli shift which are discontinuity points for Lyapunov exponents in the Hölder 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 Hölder continuous cocycles. Our examples give evidence that this theorem is optimal within certain families of Hölder cocycles.

16 pages. v2: Modified the abstract. Improved exposition in the introduction. v3: Final arXiv version. Variance calculation at the end has been corrected. Many typos corrected. To appear in Ergodic Theory and Dynamical Systems

References in corpus (3)

• Sharp error terms for return time statistics under mixing conditions
• Continuity of Lyapunov Exponents for Cocycles with Invariant Holonomies
• Continuity of Lyapunov Exponents for Random 2D Matrices

Cited by in corpus (2)

• Continuity of Lyapunov exponents for non-uniformly fiber-bunched cocycles
• Continuity of Lyapunov exponents in all Hölder topologies for irreducible cocycles