Rigidity on horocycles and hypercycles

We show that a bijection f:H²² of the hyperbolic plane that sends horocycles to horocycles (respectively hypercycles to hypercycles) is an isometry. This extends a previous result of J. Jeffers on geodesics to all curves with constant curvature in H². We go beyond by showing that every abstract automorphism of the geodesic graph (respectively horocycles and hypercycles graphs) is induced by an earthquake map (respectively an isometry) of H². This shadowed the difference between the geometry of geodesics and that of horocycles/hypercycles.