2019, 15: 143-163. doi: 10.3934/jmd.2019017

Entropy and quasimorphisms

1. 

Ben Gurion University of the Negev, Beer Sheva, 8410501, Israel

2. 

University of Regensburg, 93053 Regensburg, Germany and University of Wrocław, 50-137 Wrocław, Poland

Received  May 06, 2018 Revised  February 15, 2019 Published  June 2019

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.

Citation: Michael Brandenbursky, Michał Marcinkowski. Entropy and quasimorphisms. Journal of Modern Dynamics, 2019, 15: 143-163. doi: 10.3934/jmd.2019017
References:
[1]

Travaux de Thurston Sur Les Surfaces, Séminaire Orsay, With an English summary, Astérisque, 66–67, Société Mathématique de France, Paris, 1979.  Google Scholar

[2]

A. Banyaga, The Structure of Classical Diffeomorphism Groups, Mathematics and its Applications, 400, Kluwer Academic Publishers Group, Dordrecht, 1997. doi: 10.1007/978-1-4757-6800-8.  Google Scholar

[3]

M. Bestvina and K. Fujiwara, Bounded cohomology of subgroups of mapping class groups, Geom. Topol., 6 (2002), 69–89 (electronic). doi: 10.2140/gt.2002.6.69.  Google Scholar

[4]

J. S. Birman, Mapping class groups and their relationship to braid groups, Comm. Pure Appl. Math., 22 (1969), 213-238.  doi: 10.1002/cpa.3160220206.  Google Scholar

[5]

J. S. Birman, Braids, Links, and Mapping Class Groups, Annals of Mathematics Studies, No. 82, Princeton University Press, Princeton, N.J.; University of Tokyo Press, Tokyo, 1974.  Google Scholar

[6]

R. Bowen, Entropy for group endomorphisms and homogeneous spaces, Trans. Amer. Math. Soc., 153 (1971), 401-414.  doi: 10.1090/S0002-9947-1971-0274707-X.  Google Scholar

[7]

M. Brandenbursky, On quasi-morphisms from knot and braid invariants, J. Knot Theory Ramifications, 20 (2011), 1397-1417.  doi: 10.1142/S0218216511009212.  Google Scholar

[8]

M. Brandenbursky, Bi-invariant metrics and quasi-morphisms on groups of Hamiltonian diffeomorphisms of surfaces, Internat. J. Math., 26 (2015), 1550066, 29 pages. doi: 10.1142/S0129167X15500664.  Google Scholar

[9]

M. Brandenbursky and J. Kȩdra, On the autonomous metric on the group of area-preserving diffeomorphisms of the 2-disc, Algebr. Geom. Topol., 13 (2013), 795-816.  doi: 10.2140/agt.2013.13.795.  Google Scholar

[10]

M. Brandenbursky, J. Kedra and E. Shelukhin, On the autonomous norm on the group of Hamiltonian diffeomorphisms of the torus, Comm. Contemp. Math., 20 (2018), 1750042, 27pp. doi: 10.1142/S0219199717500420.  Google Scholar

[11]

M. Brandenbursky and E. Shelukhin, On the Lp-geometry of autonomous Hamiltonian diffeomorphisms of surfaces, Math. Res. Lett., 22 (2015), 1275-1294.  doi: 10.4310/MRL.2015.v22.n5.a1.  Google Scholar

[12]

D. Burago, S. Ivanov and L. Polterovich, Conjugation-invariant norms on groups of geometric origin, in Groups of Diffeomorphisms, Adv. Stud. Pure Math., 52, Math. Soc. Japan, Tokyo, 2008,221–250. doi: 10.2969/aspm/05210221.  Google Scholar

[13]

D. Calegari, MSJ Memoirs, Vol. 20, Mathematical Society of Japan, Tokyo, 2009. doi: 10.1142/e018.  Google Scholar

[14]

E. I. Dinaburg, A connection between various entropy characterizations of dynamical systems, Izv. Akad. Nauk SSSR Ser. Mat., 35 (1971), 324-366.   Google Scholar

[15]

M. Entov, L. Polterovich and P. Py, On continuity of quasimorphisms for symplectic maps, With an appendix by Michael Khanevsky, in Perspectives in Analysis, Geometry, and Topology, Progr. Math., 296, Birkhäuser/Springer, New York, 2012,169–197. doi: 10.1007/978-0-8176-8277-4_8.  Google Scholar

[16]

J.-M. Gambaudo and E. E. Pécou, Dynamical cocycles with values in the Artin braid group, Ergodic Theory Dynam. Systems, 19 (1999), 627-641.  doi: 10.1017/S0143385799130207.  Google Scholar

[17]

J.-M. Gambaudo and É. Ghys, Commutators and diffeomorphisms of surfaces, Ergodic Theory Dynam. Systems, 24 (2004), 1591-1617.  doi: 10.1017/S0143385703000737.  Google Scholar

[18]

W. J. Harvey, Boundary structure of the modular group, in Riemann Surfaces and Related Topics: Proceedings of the 1978 Stony Brook Conference (State Univ. New York, Stony Brook, N.Y., 1978), Ann. of Math. Stud., 97, Princeton Univ. Press, Princeton, N.J., 1981,245–251.  Google Scholar

[19]

T. Ishida, Quasi-morphisms on the group of area-preserving diffeomorphisms of the 2-disk via braid groups, Proc. Amer. Math. Soc. Ser. B, 1 (2014), 43-51.  doi: 10.1090/S2330-1511-2014-00002-X.  Google Scholar

[20]

N. V. Ivanov, Subgroups of Teichmüller Modular Groups, Translated from the Russian by E. J. F. Primrose and revised by the author, Translations of Mathematical Monographs, 115, American Mathematical Society, Providence, RI, 1992.  Google Scholar

[21]

D. Margalit, Thurston's work on surfaces [book review of MR3053012], Bull. Amer. Math. Soc. (N.S.), 51 (2014), 151-161.  doi: 10.1090/S0273-0979-2013-01419-8.  Google Scholar

[22]

H. A. Masur and Y. N. Minsky, Geometry of the complex of curves. I. Hyperbolicity, Invent. Math., 138 (1999), 103-149.  doi: 10.1007/s002220050343.  Google Scholar

[23]

L. Polterovich, Floer homology, dynamics and groups, in Morse Theoretic Methods in Nonlinear Analysis and in Symplectic Topology, NATO Sci. Ser. II Math. Phys. Chem., 217, Springer, Dordrecht, 2006,417–438. doi: 10.1007/1-4020-4266-3_09.  Google Scholar

[24]

L. Polterovich and E. Shelukhin, Autonomous Hamiltonian flows, Hofer's geometry and persistence modules, Selecta Math. (N.S.), 22 (2016), 227-296.  doi: 10.1007/s00029-015-0201-2.  Google Scholar

[25]

S. Schleimer, Notes on the complex of curves, http://homepages.warwick.ac.uk/ masgar/Maths/notes.pdf. Google Scholar

[26]

T. Tsuboi, On the uniform simplicity of diffeomorphism groups, in Differential Geometry, World Sci. Publ., Hackensack, NJ, 2009, 43–55. doi: 10.1142/9789814261173_0004.  Google Scholar

[27]

T. Tsuboi, On the uniform perfectness of the groups of diffeomorphisms of even-dimensional manifolds, Comment. Math. Helv., 87 (2012), 141-185.  doi: 10.4171/CMH/251.  Google Scholar

[28]

Y. Yomdin, Volume growth and entropy, Israel J. Math., 57 (1987), 285-300.  doi: 10.1007/BF02766215.  Google Scholar

[29]

L. S. Young, Entropy of continuous flows on compact 2-manifolds, Topology, 16 (1977), 469-471.  doi: 10.1016/0040-9383(77)90053-2.  Google Scholar

show all references

References:
[1]

Travaux de Thurston Sur Les Surfaces, Séminaire Orsay, With an English summary, Astérisque, 66–67, Société Mathématique de France, Paris, 1979.  Google Scholar

[2]

A. Banyaga, The Structure of Classical Diffeomorphism Groups, Mathematics and its Applications, 400, Kluwer Academic Publishers Group, Dordrecht, 1997. doi: 10.1007/978-1-4757-6800-8.  Google Scholar

[3]

M. Bestvina and K. Fujiwara, Bounded cohomology of subgroups of mapping class groups, Geom. Topol., 6 (2002), 69–89 (electronic). doi: 10.2140/gt.2002.6.69.  Google Scholar

[4]

J. S. Birman, Mapping class groups and their relationship to braid groups, Comm. Pure Appl. Math., 22 (1969), 213-238.  doi: 10.1002/cpa.3160220206.  Google Scholar

[5]

J. S. Birman, Braids, Links, and Mapping Class Groups, Annals of Mathematics Studies, No. 82, Princeton University Press, Princeton, N.J.; University of Tokyo Press, Tokyo, 1974.  Google Scholar

[6]

R. Bowen, Entropy for group endomorphisms and homogeneous spaces, Trans. Amer. Math. Soc., 153 (1971), 401-414.  doi: 10.1090/S0002-9947-1971-0274707-X.  Google Scholar

[7]

M. Brandenbursky, On quasi-morphisms from knot and braid invariants, J. Knot Theory Ramifications, 20 (2011), 1397-1417.  doi: 10.1142/S0218216511009212.  Google Scholar

[8]

M. Brandenbursky, Bi-invariant metrics and quasi-morphisms on groups of Hamiltonian diffeomorphisms of surfaces, Internat. J. Math., 26 (2015), 1550066, 29 pages. doi: 10.1142/S0129167X15500664.  Google Scholar

[9]

M. Brandenbursky and J. Kȩdra, On the autonomous metric on the group of area-preserving diffeomorphisms of the 2-disc, Algebr. Geom. Topol., 13 (2013), 795-816.  doi: 10.2140/agt.2013.13.795.  Google Scholar

[10]

M. Brandenbursky, J. Kedra and E. Shelukhin, On the autonomous norm on the group of Hamiltonian diffeomorphisms of the torus, Comm. Contemp. Math., 20 (2018), 1750042, 27pp. doi: 10.1142/S0219199717500420.  Google Scholar

[11]

M. Brandenbursky and E. Shelukhin, On the Lp-geometry of autonomous Hamiltonian diffeomorphisms of surfaces, Math. Res. Lett., 22 (2015), 1275-1294.  doi: 10.4310/MRL.2015.v22.n5.a1.  Google Scholar

[12]

D. Burago, S. Ivanov and L. Polterovich, Conjugation-invariant norms on groups of geometric origin, in Groups of Diffeomorphisms, Adv. Stud. Pure Math., 52, Math. Soc. Japan, Tokyo, 2008,221–250. doi: 10.2969/aspm/05210221.  Google Scholar

[13]

D. Calegari, MSJ Memoirs, Vol. 20, Mathematical Society of Japan, Tokyo, 2009. doi: 10.1142/e018.  Google Scholar

[14]

E. I. Dinaburg, A connection between various entropy characterizations of dynamical systems, Izv. Akad. Nauk SSSR Ser. Mat., 35 (1971), 324-366.   Google Scholar

[15]

M. Entov, L. Polterovich and P. Py, On continuity of quasimorphisms for symplectic maps, With an appendix by Michael Khanevsky, in Perspectives in Analysis, Geometry, and Topology, Progr. Math., 296, Birkhäuser/Springer, New York, 2012,169–197. doi: 10.1007/978-0-8176-8277-4_8.  Google Scholar

[16]

J.-M. Gambaudo and E. E. Pécou, Dynamical cocycles with values in the Artin braid group, Ergodic Theory Dynam. Systems, 19 (1999), 627-641.  doi: 10.1017/S0143385799130207.  Google Scholar

[17]

J.-M. Gambaudo and É. Ghys, Commutators and diffeomorphisms of surfaces, Ergodic Theory Dynam. Systems, 24 (2004), 1591-1617.  doi: 10.1017/S0143385703000737.  Google Scholar

[18]

W. J. Harvey, Boundary structure of the modular group, in Riemann Surfaces and Related Topics: Proceedings of the 1978 Stony Brook Conference (State Univ. New York, Stony Brook, N.Y., 1978), Ann. of Math. Stud., 97, Princeton Univ. Press, Princeton, N.J., 1981,245–251.  Google Scholar

[19]

T. Ishida, Quasi-morphisms on the group of area-preserving diffeomorphisms of the 2-disk via braid groups, Proc. Amer. Math. Soc. Ser. B, 1 (2014), 43-51.  doi: 10.1090/S2330-1511-2014-00002-X.  Google Scholar

[20]

N. V. Ivanov, Subgroups of Teichmüller Modular Groups, Translated from the Russian by E. J. F. Primrose and revised by the author, Translations of Mathematical Monographs, 115, American Mathematical Society, Providence, RI, 1992.  Google Scholar

[21]

D. Margalit, Thurston's work on surfaces [book review of MR3053012], Bull. Amer. Math. Soc. (N.S.), 51 (2014), 151-161.  doi: 10.1090/S0273-0979-2013-01419-8.  Google Scholar

[22]

H. A. Masur and Y. N. Minsky, Geometry of the complex of curves. I. Hyperbolicity, Invent. Math., 138 (1999), 103-149.  doi: 10.1007/s002220050343.  Google Scholar

[23]

L. Polterovich, Floer homology, dynamics and groups, in Morse Theoretic Methods in Nonlinear Analysis and in Symplectic Topology, NATO Sci. Ser. II Math. Phys. Chem., 217, Springer, Dordrecht, 2006,417–438. doi: 10.1007/1-4020-4266-3_09.  Google Scholar

[24]

L. Polterovich and E. Shelukhin, Autonomous Hamiltonian flows, Hofer's geometry and persistence modules, Selecta Math. (N.S.), 22 (2016), 227-296.  doi: 10.1007/s00029-015-0201-2.  Google Scholar

[25]

S. Schleimer, Notes on the complex of curves, http://homepages.warwick.ac.uk/ masgar/Maths/notes.pdf. Google Scholar

[26]

T. Tsuboi, On the uniform simplicity of diffeomorphism groups, in Differential Geometry, World Sci. Publ., Hackensack, NJ, 2009, 43–55. doi: 10.1142/9789814261173_0004.  Google Scholar

[27]

T. Tsuboi, On the uniform perfectness of the groups of diffeomorphisms of even-dimensional manifolds, Comment. Math. Helv., 87 (2012), 141-185.  doi: 10.4171/CMH/251.  Google Scholar

[28]

Y. Yomdin, Volume growth and entropy, Israel J. Math., 57 (1987), 285-300.  doi: 10.1007/BF02766215.  Google Scholar

[29]

L. S. Young, Entropy of continuous flows on compact 2-manifolds, Topology, 16 (1977), 469-471.  doi: 10.1016/0040-9383(77)90053-2.  Google Scholar

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