HISTORIC WANDERING DOMAINS NEAR CYCLES

The paper’s Theorem A states exactly the main claim the candidate advances: in a Cr-Newhouse domain N (r ≥ 1) of homoclinic tangencies associated with saddles satisfying condition (1) (complex pair λe±iφ, |λ^2 γ| < 1 < |λ γ|), there is a dense subset of diffeomorphisms with non-trivial historic contractive wandering domains, and such Newhouse domains arise near equidimensional and certain 3D heterodimensional cycles; see the statement of Theorem A and its surrounding discussion , together with the definition of condition (1) . The proof sketched in §2.1 matches the candidate’s Steps 1, 3, and 4: a GST-type rescaling provides a two-dimensional normally hyperbolic attracting invariant manifold Mn on which the first return is Hénon-like and has a dissipative homoclinic tangency (Lemma 2.1) ; then applying Kiriki–Soma on the 2D restriction and thickening along strong-stable plaques yields ambient historic contractive wandering domains densely in N . The “Moreover” proximity statements to cycles are also aligned with the paper’s arguments (equidimensional cycles with (H1)–(H2) and blender condition (H3) in C1; and the 3D heterodimensional cycle case via approximation to equidimensional cycles and Tatjer tangencies) . However, the candidate’s Step 2 incorrectly asserts that a Tatjer/Denjoy route in C1 already produces historic wandering domains on the 2D restriction. The paper explicitly emphasizes that known high-dimensional Denjoy-type wandering domains (including KNS17) are not historic; historicity is recovered by reducing to a 2D dissipative homoclinic tangency where the Kiriki–Soma surface theorem (C2) applies, using smooth maps inside the Newhouse domain or via the Tatjer-to-2D reduction (Theorems B and C) , , . Therefore, while the core outline matches the paper, the model conflates “wandering” with “historic” in the C1/Tatjer/Denjoy step.