On the Lp-geometry of autonomous Hamiltonian diffeomorphisms of surfaces

We prove a number of results on the interrelation between the L-p-metric on the group of Hamiltonian diffeomorphisms of surfaces and the subset A of autonomous Hamiltonian diffeomorphisms. More precisely, we show that there are Hamiltonian diffeomorphisms of all surfaces of genus g >= 2 or g = 0 lying arbitrarily L-p-far from the subset A, answering a variant of a question of Polterovich for the L-p-metric.