ON LIMIT SETS FOR GEODESICS OF MEROMORPHIC CONNECTIONS

The paper constructs three families of examples and explains their dynamics using the branched-affine/flat-structure viewpoint: (i) hyperbolic cylinders on CP^1 for Fuchsian meromorphic connections, including self-intersecting cylinders with limit cycles; trajectories entering such cylinders accumulate on closed geodesics, possibly self-intersecting (sections 3.1–3.2) . (ii) On CP^1 with k-differentials, the authors give explicit quartic and cubic examples where geodesics have infinitely many self-intersections yet are not dense, via a slit-gluing construction (section 4.2) . (iii) On a torus with a quartic differential, they build an invariant component whose ω-limit set is the complement of a square; dynamics there is given by an irrational IET, hence minimal, and trajectories have infinitely many self-intersections (section 4.4) . The model’s Part (A) and (C) align with the paper’s constructions. However, Part (B) incorrectly asserts that a quartic differential with four double poles has ring domains of closed trajectories near each pole. In the flat geometry of k-differentials, a singularity of order a > −k yields a conical angle (a+k)2π/k; for k=4 and a=−2 one gets an angle π, not a ring domain, contradicting the model’s barrier argument (section 4.1) . Thus the model’s CP^1 quartic example and its non-density proof are flawed, while the paper’s examples are coherent and supported by the stated framework.