Typical coexistence of infinitely many strange attractors

The paper states and proves Theorem A: in a Cd,r-Berger domain U (m ≥ 3, 3 ≤ d < r − 1), there is a residual set R of k-parameter families for which, for Lebesgue-a.e. parameter, the diffeomorphism has infinitely many non-hyperbolic strange attractors (Theorem A and its reduction to Theorem B) . The proof skeleton is: (i) parameter-dependent renormalization producing Hénon-like families (Definition 2.1, Proposition 2.2, Lemma 2.4, and the rescaling to the parabola family) ; (ii) use of prevalence results for Hénon-like families to get a uniform positive-measure set of good parameters in each window; and (iii) a Borel–Cantelli argument across infinitely many windows to conclude that a.e. parameter falls in infinitely many good windows (the “second Borel–Cantelli Lemma” step) . The candidate solution follows the same blueprint: persistent tangencies in Berger domains, Palis–Takens/Mora–Viana renormalization to small-Jacobian Hénon-like maps, a uniform positive-measure parameter set in each unfolding window, and a Kolmogorov-typicality scheme using a quasi-independence/Borel–Cantelli mechanism. However, both accounts are incomplete on the probabilistic step: the paper invokes the second Borel–Cantelli lemma without establishing independence or a suitable quasi-independence estimate for the parameter events across different windows (the argument extracts disjoint subcollections within each level but does not control cross-level correlations) . The candidate solution mentions quasi-independence but does not supply the needed estimate either. Aside from this gap, the renormalization and prevalence parts match and are consistent with the cited framework (Definition 1.1; Theorem B; Definition 2.1; Lemma 2.4) .