A distance between orbits that controls commutator estimates and invertibility of operators

A distance between orbit spaces generated by a single element is introduced and it is shown that if an operator is invertible in one orbit it is also invertible in nearby orbits, thus proving a version of Shneiberg's theorem for orbital methods. The same machinery is used to extend the celebrated Rochberg-Weiss commutator theorem to the setting of orbital methods. It is shown that these results apply to the real and complex methods of interpolation by proving that these methods can be suitably obtained as orbits of a single element.