# American Institute of Mathematical Sciences

2019, 15: 143-163. doi: 10.3934/jmd.2019017

## Entropy and quasimorphisms

 1 Ben Gurion University of the Negev, Beer Sheva, 8410501, Israel 2 University of Regensburg, 93053 Regensburg, Germany and University of Wrocław, 50-137 Wrocław, Poland

Received  May 06, 2018 Revised  February 15, 2019 Published  June 2019

Let $S$ be a compact oriented surface. We construct homogeneous quasimorphisms on ${\rm Diff}(S, \operatorname{area})$, on ${\rm Diff}_0(S, \operatorname{area})$, and on ${\rm Ham}(S)$, generalizing the constructions of Gambaudo-Ghys and Polterovich.

We prove that there are infinitely many linearly independent homogeneous quasimorphisms on ${\rm Diff}(S, \operatorname{area})$, on ${\rm Diff}_0(S, \operatorname{area})$, and on ${\rm Ham}(S)$ whose absolute values bound from below the topological entropy. In cases when $S$ has a positive genus, the quasimorphisms we construct on ${\rm Ham}(S)$ are $C^0$-continuous.

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 on ${\rm Ham}(S)$ is unbounded.

Citation: Michael Brandenbursky, Michał Marcinkowski. Entropy and quasimorphisms. Journal of Modern Dynamics, 2019, 15: 143-163. doi: 10.3934/jmd.2019017
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Loop $\gamma$ and arcs $a$ and $b$