On the $L^p$-geometry of autonomous Hamiltonian diffeomorphisms of surfaces

TL;DR

This paper investigates the geometric properties of the group of Hamiltonian diffeomorphisms on surfaces under the $L^p$-metric, revealing that many such diffeomorphisms are arbitrarily distant from autonomous ones, especially on surfaces of genus not equal to one.

Contribution

It demonstrates that for surfaces of genus not equal to one, there exist Hamiltonian diffeomorphisms arbitrarily far from autonomous diffeomorphisms in the $L^p$-metric, addressing a question posed by Polterovich.

Findings

Existence of Hamiltonian diffeomorphisms arbitrarily $L^p$-far from autonomous ones.

Results apply to all surfaces with genus $g eq 1$.

Provides insights into the $L^p$-geometry of Hamiltonian diffeomorphism groups.

Abstract

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 of all autonomous Hamiltonian diffeomorphisms. More precisely, we show that there are Hamiltonian diffeomorphisms of all surfaces of genus $g\neq 1$ lying arbitrarily $L^p$-far from this subset; answering a variant of a question of Polterovich for the $L^p$-metric.