Some Generic Properties of Partially Hyperbolic Endomorphisms

The paper’s Theorem A states the exact dichotomy at issue: for a transitive partially hyperbolic endomorphism and each σ in {u,c}, either f is σ-special or there is a residual set on which every point has infinitely many σ-directions, see the statement and surrounding discussion of Theorem A . The paper’s proof proceeds via Proposition 3.1 (a Baire-category/angle-contraction argument using continuity of the bundles on the natural extension and monotonicity along forward orbits) to first find a point with σ(x)=∞, and then uses transitivity to spread this to a residual set . The candidate solution proves the same dichotomy but with a constructive argument on the inverse-limit space M_f: it builds cylinder neighborhoods giving two distinct σ-directions over a dense open set and then, using returns of x to that set, splices prehistories with disjoint negative-time windows to produce infinitely many distinct σ-directions at a residual set of points. This relies on continuity of the invariant σ-bundles on M_f (as in Proposition 2.8) and standard local-inverse charts for a local diffeomorphism, which the paper also establishes . Thus both are correct; the paper uses a Baire/angle-contraction route (via Proposition 2.5) while the model uses a natural-extension/cylinder-splicing construction .