Continuum-wise hyperbolicity is exactly the pseudo-Anosov dynamics with spine singularities

Abstract

We establish a complete structural classification for cw${}_F$-hyperbolic surface homeomorphisms. Specifically, we prove that a surface homeomorphism is cw${}_F$-hyperbolic if, and only if, it is a pseudo-Anosov homeomorphism whose singularities consist exclusively of spines (1-prongs). Furthermore, we classify these systems up to topological conjugacy, showing that every such homeomorphism is conjugate to either an Anosov automorphism on the torus ${\mathbb{T}}^2$ or to a standard hyperelliptic quotient on ${\mathbb{S}}^2$ of a hyperbolic affine toral map. As a rigid consequence of this classification, we show that such dynamics are strictly obstructed on surfaces of genus greater than one, the Klein bottle, and the projective plane.