ABUNDANCE OF OBSERVABLE LYAPUNOV IRREGULAR SETS

The paper establishes Theorem A: for some g in a Newhouse open set (r≥2), every sufficiently small neighborhood O of g contains an uncountable family L of diffeomorphisms that are pairwise non-topologically-conjugate; for each f in L there are open sets U_f and cone V_f such that every x in U_f is Lyapunov-irregular for all nonzero v in V_f, and L splits into uncountable subfamilies R (Birkhoff-regular on U_f) and I (Birkhoff-irregular on U_f). This statement and its proof are presented explicitly and in detail, including the Colli–Vargas return-map setup, the control of higher-order terms, and the construction of the uncountable, non-conjugate families with the Birkhoff dichotomy (see Theorem A and Section 4, including Theorem 4.8 and the concluding argument) . By contrast, the candidate solution hinges on a stronger, unproven premise: it assumes as input that within any Newhouse open set there is a dense subset of maps already exhibiting the open-set Lyapunov-irregular property with a Birkhoff regular/irregular dichotomy, attributed to the same 2021 paper. The paper itself does not prove that dense-subset statement; rather, it proves the existence of a specific g with the desired neighborhood-universal property and remarks that a dense-set extension is plausible, not established. Consequently, the model’s Step (A) misstates the available result and uses it critically to select f_R and f_I in an arbitrary small neighborhood O; this logical gap undermines the model’s argument even though its later use of periodic-spot technology (Turaev) could, in principle, supply a different route to non-conjugacy. In short: the paper’s argument is correct and complete for the stated theorem, while the model’s solution relies on a non-cited, unproven strengthening.