TY - JOUR
T1 - Entropy and quasimorphisms
AU - Brandenbursky, Michael
AU - Marcinkowski, Michał
N1 - Funding Information:
We would like to thank Mladen Bestvina, Danny Calegari, and Benson Farb for fruitful conversations. We would like also to thank Bernd Ammann and Ulrich Bunke for spotting an inaccuracy in the earlier proof of Lemma 2.1. Both authors were partially supported by GIF-Young Grant number I-2419-304.6/2016 and by SFB 1085 “Higher Invariants” funded by DFG. Part of this work has been done during the first author’s stay at the University of Regensburg and the second author’s stay at the Ben Gurion University. We wish to express our gratitude to both places for the support and excellent working conditions.
Funding Information:
Both authors were partially supported by GIF-Young Grant number I-2419-304.6/2016 and by SFB 1085 “Higher Invariants” funded by DFG. Part of this work has been done during the first author’s stay at the University of Regensburg and the second author’s stay at the Ben Gurion University. We wish to express our gratitude to both places for the support and excellent working conditions.
Publisher Copyright:
© 2019 AIMSCIENCES.
PY - 2019/1/1
Y1 - 2019/1/1
N2 - 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.
AB - 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.
KW - Area-preserving and Hamiltonian diffeomorphisms
KW - Braid groups
KW - Conjugation-invariant norms
KW - Mapping class groups
KW - Quasimorphisms
KW - Topological entropy
UR - http://www.scopus.com/inward/record.url?scp=85073460502&partnerID=8YFLogxK
U2 - 10.3934/jmd.2019017
DO - 10.3934/jmd.2019017
M3 - Article
AN - SCOPUS:85073460502
SN - 1930-5311
VL - 15
SP - 143
EP - 163
JO - Journal of Modern Dynamics
JF - Journal of Modern Dynamics
ER -