Upper bound on the rate of mixing for the Earthquake flow on moduli spaces

The paper proves that any uniform polynomial mixing bound for the earthquake flow on P^1 M_g must have exponent d ≤ 6g−5, using test functions supported where the systole is small together with Minsky–Weiss nondivergence and explicit Lipschitz and volume estimates; see Theorem 1.2 and the end of Section 3 (inequality (2) leading to µ^{6g−5} ≤ const·µ^d) . The candidate solution independently reaches the same exponent by building Lipschitz bump functions on the lamination factor, exploiting that the earthquake flow fixes the lamination coordinate, and comparing the covariance lower bound (∼δ^{6g−7}) with the Lipschitz norms (∼δ^{-1}) to obtain the obstruction δ^{6g−5} ≲ t^{−d}. The logic aligns with the paper’s conclusion, though a minor technical point in the candidate’s write-up needs stating (uniform control of the small-ball Thurston mass or using a mean-zero bump to avoid the mean term). Overall, both arguments correctly force d ≤ 6g−5; the proofs are different in construction and tools, but consistent in outcome .