CONTINUUM-WISE HYPERBOLICITY AND PERIODIC POINTS

The paper proves (i) a periodic closing/shadowing theorem for cw-hyperbolic homeomorphisms with jointly continuous holonomies (Theorem 2.7) and (ii) periodic specification under mixing (Theorem 2.12), using a self-similar hyperbolic cw-metric and jointly pseudo-isometric holonomies to control holonomy rectangles and a careful r-block construction to close periodic pseudo-orbits . The candidate solution mirrors the rectangle scheme and the self-similar/pseudo-isometric reduction, but commits a critical error: it uses cw-expansiveness to claim uniqueness of shadowing (and hence f^n(p)=p) when 2α<c. That uniqueness is false in the cw-expansive setting (even periodic pseudo-orbits can be shadowed by non-periodic points), a nuance explicitly noted in the paper’s discussion of shadowing multiplicity in the cw-framework and addressed in the proof via the r-iterate closing argument rather than expansivity-style uniqueness. The paper’s arguments are internally coherent with the needed constants and lemmas (e.g., the jointly pseudo-isometric holonomies after self-similarization) , while the model’s proof hinges on an invalid uniqueness step.