Article Contents
Article Contents

# Entropy and quasimorphisms

• 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.

Mathematics Subject Classification: Primary: 57S05; Secondary: 20F65, 37B40.

 Citation:

• Figure 3.1.  Loop $\gamma$ and arcs $a$ and $b$

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