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Entropy and quasimorphisms

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  • 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.


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

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