Semi-continuity of Oseledets subspaces and Pesin sets with exponentially small tails

The paper establishes two results: (i) for Axiom A basic sets, if the Oseledets subspaces Ei(x) depend upper semi‑continuously, then there exists a Pesin set with exponentially small tails (Theorem 1), and (ii) for open billiard flows, the time‑one map’s Oseledets subspaces depend upper semi‑continuously for any Gibbs measure (Theorem 2), yielding the same Pesin sets as a corollary. The statements, definitions, and proof structure are explicit in the PDF, including the definition of Pesin sets with exponentially small tails and the upper semi‑continuity condition (Definition 2) and the main theorems (Theorems 1 and 2) . The proof of Theorem 1 is via a symbolic (SFT) reduction and a general large‑deviations theorem (Theorem 4) implying exponential tails, combined with a result from Gouëzel–Stoyanov to obtain Pesin sets with exponentially small tails . Theorem 2 is proved by careful billiard derivative estimates (Theorem 5) and a geometric argument . The candidate solution reaches the same conclusions by a different route: Part A uses a reduction to SFT and invokes Stoyanov’s 2022 result, but it misstates the hypothesis as upper semi‑continuity of flags rather than subspaces; nevertheless, since upper semi‑continuity of Ei implies upper semi‑continuity of the partial sums Fj, their route is compatible with the actual hypotheses. Part B appeals to Araújo–Bufetov–Filip (2016) to obtain Hölder continuity of Oseledets subspaces on large‑measure compact sets for the derivative cocycle over the coded SFT, then upgrades to a full‑measure set with upper semi‑continuity; this is an alternative to the paper’s geometric proof. With minor corrections (fixing the flagged mis‑citation and checking standard cocycle regularity hypotheses), the candidate’s approach is sound and yields the same results.