On recurrence in zero-dimensional locally compact flow with compactly generated phase group

The paper proves the nine-way equivalence (Theorem 4a) by a closed cycle of implications that does not require upgrading pcontinuity to upper semicontinuity nor deriving (9) from (3); crucial steps are (2)⇒(1), (3)⇒(2), (1)⇒(8), (8)⇒(7), (7)⇒(6)⇒(5)⇒(4)⇒(3), plus (7)⇒(9) and (9)⇒(3) to tie in (9) . By contrast, the model asserts two nonstandard and generally false implications in this noncompact setting: (i) pcontinuity ⇒ upper semicontinuity of the orbit-closure map, despite the paper’s explicit warning that Ro closed does not imply upper semicontinuity in general ; and (ii) local weak almost periodicity ⇒ upper semicontinuity. These missteps are unnecessary for the equivalence but render the model’s proof logically incorrect as written. The model also claims compact-flow equivalences “carry over” wholesale to the locally compact setting; the paper avoids this by using the (1)⇒(8) net/cone argument and the (8)⇒(7) reduction via compact-open neighborhoods , .