STRONGLY DISTRIBUTIONAL CHAOS OF IRREGULAR ORBITS THAT ARE NOT UNIFORMLY HYPERBOLIC

The paper establishes the result via an abstract, verifiable framework (nested transitive locally maximal hyperbolic sets with exponential shadowing/expansiveness) and a precise α-DC1 construction that is uniform in t, culminating in Theorems A, E, and F. The candidate solution sketches a different, plausible program (finite gluing via λ-lemma and two periodic ‘building blocks’), but it has at least two critical gaps: (1) its α-DC1 limsup estimate depends on a fixed t in the stage-size choice and does not provide a uniform (all t > 0) argument required by the definition; (2) it relies on visits to arbitrarily small stable/unstable angles inside a non-uniformly hyperbolic homoclinic class/support without justification (this need not hold absent genericity). It also assumes uniformly bounded heteroclinic transition times inside S_μ that are not established. Consequently, the paper’s argument is correct and complete within its framework, whereas the model’s proof is incomplete.