PSEUDO-ARC AS THE ATTRACTOR IN THE DISC - TOPOLOGICAL AND MEASURE-THEORETIC ASPECTS

The paper proves that for a residual T ⊂ C_λ(I), every f ∈ T has arbitrarily small crooked iterates (Theorem 1.1), which via a standard crookedness criterion implies the inverse limit lim←(I,f) is the pseudo-arc (Lemma 3.1), and then builds a parametrized Brown–Barge–Martin (BBM) realization on the disk yielding items (a)–(f): conjugacy to the natural extension, Hausdorff continuity of attractors, existence and uniqueness-in-density of physical measures with controlled basins, transitivity and shadowing, and statistical stability (Theorem 1.6). The candidate solution follows the same chain: (1) generic crookedness, (2) pseudo-arc inverse limits, (3) intersect with generic leo/weak-mixing/shadowing sets, (4) continuous BBM realization, (5) construction of physical measures and full-basin cases, (6) transitivity/shadowing via properties passing to inverse limits, and (7) weak* continuity of the measure. All these correspond closely to the paper’s statements and structure, with only minor bibliographic imprecision. Key points are directly stated in the paper: Theorem 1.1 (generic crooked iterates) , Lemma 3.1 (crookedness ⇒ pseudo-arc) , and Theorem 1.6 (a)–(f) (parametrized BBM with physical measures and statistical stability) . Measure-theoretic set-up and OU background are also provided in Section 5 , and generic leo/weak-mixing/shadowing come from cited prior works summarized in the paper’s introduction and Corollary 1.4 .