On ball expanding maps

The uploaded paper’s abstract states that for ball expanding maps the set of periodic points is dense in the chain recurrent set, but not equal to it (item (1) in the abstract) . In the body, however, Theorem 1.1 claims the much stronger equality CR(f)=Per(f), deduced via an iterate having a Lipschitz shadowing constant <1/2 and the identities CR(f^i)=CR(f), Per(f^i)=Per(f) . This conflicts with the paper’s own examples: the tent map on [0,1] and the doubling map on S^1 are stated to be ball expanding and locally eventually onto (hence mixing) . By the paper’s Remark 1.7, mixing implies chain mixing, and with chain mixing every point is chain recurrent, so CR(f)=X in these examples . Since not every point is periodic for these maps, equality CR(f)=Per(f) is false. The candidate solution correctly flags this and replaces (1) with the sharp statement that Per(f) is dense in CR(f); the remaining items (2)–(5) agree with the paper’s (correct) statements and supporting lemmas (equivalences for zero entropy, finiteness of chain components, positivity of entropy on perfect spaces, and LEO on connected spaces) . Caveat: the model’s sketch briefly overclaims that ball expanding maps always have CR(f)=X; this is incorrect in general (see the paper’s Example 2.1 where CR(f) = {0,1} ≠ X), but it does not affect the model’s main correction to (1) nor the truth of (2)–(5) as stated in the paper .