Horocyclic Trajectories in Hyperbolic Solenoidal Surfaces of Finite Type

The paper proves the dichotomy rigorously (Theorem 2): for Y = Γ \ (PSL(2,R) × G) arising from a hyperbolic McCord solenoid, every horocycle orbit either projects to a periodic base orbit or is dense in Y, via a three-step argument using unique ergodicity of dense subgroup actions on G, Furstenberg’s equivalence, and Moore’s ergodicity to obtain a dense hR-orbit and then propagate density across fibers and to any point whose projection is nonperiodic . The model’s topological proof attempts to “synchronize” base density with fiber density by prescribing elements δm ∈ Γ while choosing return times tm, but its key step—asserting that pr(u0 n_{tm}) ∈ pr(δmB) implies u0 n_{tm} ∈ δmB—is false in general; one only gets u0 n_{tm} ∈ γ δm B for some γ ∈ Γ. Without an additional synchronization or measure-equidistribution input, the model’s argument does not conclude density. The McCord/principal bundle setup and the dense image ρ(Γ) ⊂ G are correctly stated (and used in the paper) , but the model omits the ergodic step that the paper leverages to bridge this gap.