In this talk I will show that gauge-invariant derivatives of scalar fields grow asymptotically along the event horizon of extremal black holes. This growth originates from conservation laws on such horizons. The resulted conserved quantities can be seen as a natural generalization of the Newman-Penrose constants along null infinity. I will also present recent generalizations by Reall and Lucietti concerning electromagnetic and linearized gravitational perturbations of extremal black holes.