Mathematical Research Letters

Volume 22 (2015)

Number 5

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

Pages: 1275 – 1294



Michael Brandenbursky (Department of Mathematics, Ben-Gurion University, Beer-Sheva, Israel)

Egor Shelukhin (Institute for Advanced Study, Princeton New Jersey, U.S.A.)


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 $\mathcal{A}$ of autonomous Hamiltonian diffeomorphisms. More precisely, we show that there are Hamiltonian diffeomorphisms of all surfaces of genus $g \geq 2$ or $g = 0$ lying arbitrarily $L^p$-far from the subset $\mathcal{A}$, answering a variant of a question of Polterovich for the $L^p$-metric.


groups of Hamiltonian diffeomorphisms, braid groups, mapping class groups, quasi-morphisms, $L^p$-metrics

2010 Mathematics Subject Classification

Primary 53-xx. Secondary 57-xx.

Accepted 2 April 2015

Published 13 April 2016