Entropy and quasimorphisms

Michael Brandenbursky, Michał Marcinkowski

Research output: Contribution to journalArticlepeer-review

7 Scopus citations


Let S be a compact oriented surface. We construct homogeneous quasimorphisms on Diff(S,area), on Diff0 (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 Diff0 (S, area), and on Ham(S) whose absolute values bound from below the topological entropy. In cases when S has a positive genus, the quasimorphisms we construct on Ham(S) are C0-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.

Original languageEnglish
Pages (from-to)143-163
Number of pages21
JournalJournal of Modern Dynamics
StatePublished - 1 Jan 2019


  • Area-preserving and Hamiltonian diffeomorphisms
  • Braid groups
  • Conjugation-invariant norms
  • Mapping class groups
  • Quasimorphisms
  • Topological entropy

ASJC Scopus subject areas

  • Analysis
  • Algebra and Number Theory
  • Applied Mathematics


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