HETERODIMENSIONAL CYCLES OF HYPERBOLIC ERGODIC MEASURES.

The paper proves Theorem 2.5 by constructing quasi-hyperbolic pseudo-orbits inside the partially hyperbolic set Λ and applying Gan’s shadowing lemma to obtain periodic orbits whose Birkhoff averages converge to any convex combination sµ+ + (1−s)µ−, with supports remaining in an arbitrarily small neighborhood of Λ. This is explicitly laid out in Section 3 (definition of the setting, construction, and completion of the proof) and culminates in Theorem 2.5. By contrast, the candidate solution assumes uniformly bounded-time true-orbit transitions U+ → U− and back based solely on the existence of transverse intersections W^u ⋔ W^ss and W^uu ⋔ W^s in Definition 2.3; this confuses heteroclinic manifold intersections with actual iterate-hitting properties of individual orbits and is not implied by the paper’s hypotheses. The candidate’s Conley–Moser return-map construction therefore lacks a justified return mechanism and uniform transition times, so the argument does not go through under the stated assumptions.