Джойнинги и типичные расширения динамических систем

The paper proves that the del Junco–Rudolph (JR) property is stable (typical) in the space of extensions Ext(S) via a two-step mechanism: (i) if S has JR and an extension R is relatively weakly mixing, then any failure of JR produces a nontrivial measurable field M of fiber-joinings solving an invariance equation (1) over the base, and (ii) typical extensions have an (I,Θ)-cocycle that forbids such fields, thereby forcing JR to lift to R, yielding Theorem 1.1 (JR is stable) . By contrast, the model’s Step 2 asserts that relative weak mixing alone suffices to lift JR from the base to the extension; this crucial implication is neither established in the paper nor obviously true, as the paper explicitly inserts the extra (I,Θ)-cocycle hypothesis to rule out the nontrivial stationary fiber joinings produced under mere relative weak mixing . The model’s conclusion matches the paper’s result, but its argument omits the essential (I,Θ)-cocycle ingredient and therefore is incomplete.