ON SOME PROPERTIES OF BASIC SETS

The paper proves—in full detail—that for any invertible zero-dimensional system with dense aperiodic points there exists a minimal continuously decisive basic set B meeting each minimal set M in exactly one point (Theorem 5.12), building the construction via the extremal set inf_f and then refining to a minimal basic/quasi-section; it also establishes the quasi-section/minimal-set equivalence (Proposition 3.2) and decisiveness of minimal quasi-sections under dense aperiodicity (Proposition 3.11). The model’s solution correctly identifies the key equivalence and cites the same existence theorem to obtain B with properties (i)–(iv). However, it contains two issues: (1) it incorrectly claims dense aperiodicity is equivalent to having no periodic points; and (2) its optional “direct” argument for empty interior from (iii) is not valid without additional hypotheses. Despite these flaws, the core reasoning aligns with the paper’s result, albeit by citation rather than by reproducing the paper’s construction, hence ‘both correct (different proofs)’.