Topological entropy and quasimorphisms

Michael Brandenbursky, Michał Marcinkowski

Published 2017-07-19Version 1

Let $D^2$ be a unit disc in the Euclidean plane and let $\Sigma_g$ be a closed hyperbolic surface of genus $g$. Denote by $Ham(D^2)$ and $Ham(\Sigma_g)$ their groups of Hamiltonian diffeomorphisms respectively. In both cases, we prove that there are infinitely many linearly independent homogeneous quasimorphisms on these groups whose absolute values bound from below the topological entropy. This result holds in case of the groups $Diff_0(\Sigma_g, area)$ and $Diff_0(D^2, area)$ as well. In addition, we define a bi-invariant metric on these groups, called the entropy metric, and show that it is unbounded. In particular, we reprove the fact that the autonomous metric is unbounded on these groups.