A Sequential Birkhoff Theorem for Slow Logarithmic Drift in Non-Uniformly Expanding Unimodal Maps

We prove a sequential (non-autonomous) Birkhoff ergodic theorem for compositions of unimodal maps fₔ䂸 (x) = 1 − uₙ x² whose parameters drift slowly to a Misiurewicz value uc. Working under an explicit uniform-inducing-scheme hypothesis — a uniform Young tower with uniform Lasota–Yorke constants and Keller–Liverani spectral stability across the relevant parameter set — we show that for every bounded-variation observable, the time average along the sequential orbit converges in mean (indeed in L², hence in probability) to the spatial average against the unique absolutely continuous invariant measure μₔ₂ of the limiting autonomous map, whenever the drift tail sup₊≥₍|uₖ − uc| tends to zero. For the logarithmic schedule |uₙ − uc| ≤ c (log n) ^−β we further obtain almost-everywhere convergence when β > 1, the threshold at which the second-moment estimate becomes summable along a ratio-one subsequence. The proof passes to the induced first-return map, where the dynamics is uniformly expanding, and combines a uniform spectral gap with a Cesàro estimate of the accumulated drift error. The result supplies the ergodic foundation — convergence of symbolic cylinder frequencies to the band-merging statistics — for non-autonomous symbolic models of arithmetic sequences; we delimit precisely what it does and does not yield for the associated density envelopes.