Entropy and quasimorphisms

TL;DR

This paper constructs new homogeneous quasimorphisms on surface diffeomorphism groups that relate to topological entropy, introduces an entropy metric, and demonstrates its unboundedness, extending previous work in symplectic topology.

Contribution

It introduces a family of homogeneous quasimorphisms on surface diffeomorphism groups that bound topological entropy and defines an unbounded entropy metric.

Findings

Existence of infinitely many linearly independent quasimorphisms bounding entropy.

The entropy metric on these groups is unbounded.

Constructed quasimorphisms are $C^0$-continuous when the surface has positive genus.

Abstract

Let $S$ be a compact oriented surface. We construct homogeneous quasimorphisms on $Diff(S, area)$, on $Diff_0(S, area)$ and on $Ham(S)$ generalizing the constructions of Gambaudo-Ghys and Polterovich. We prove that there are infinitely many linearly independent homogeneous quasimorphisms on $Diff(S, area)$, on $Diff_0(S, area)$ and on $Ham(S)$ whose absolute values bound from below the topological entropy. In case when $S$ has a positive genus, the quasimorphisms we construct on $Ham(S)$ 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 $Ham(S)$ is unbounded.