# American Institute of Mathematical Sciences

May  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. [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. [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. [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. [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. [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. [7] M. Brandenbursky, On quasi-morphisms from knot and braid invariants, J. Knot Theory Ramifications, 20 (2011), 1397-1417. doi: 10.1142/S0218216511009212. [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. [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. [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. [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. [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. [13] D. Calegari, MSJ Memoirs, Vol. 20, Mathematical Society of Japan, Tokyo, 2009. doi: 10.1142/e018. [14] E. I. Dinaburg, A connection between various entropy characterizations of dynamical systems, Izv. Akad. Nauk SSSR Ser. Mat., 35 (1971), 324-366. [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. [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. [17] J.-M. Gambaudo and É. Ghys, Commutators and diffeomorphisms of surfaces, Ergodic Theory Dynam. Systems, 24 (2004), 1591-1617. doi: 10.1017/S0143385703000737. [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. [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. [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. [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. [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. [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. [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. [25] S. Schleimer, Notes on the complex of curves, http://homepages.warwick.ac.uk/ masgar/Maths/notes.pdf. [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. [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. [28] Y. Yomdin, Volume growth and entropy, Israel J. Math., 57 (1987), 285-300. doi: 10.1007/BF02766215. [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.

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. [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. [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. [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. [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. [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. [7] M. Brandenbursky, On quasi-morphisms from knot and braid invariants, J. Knot Theory Ramifications, 20 (2011), 1397-1417. doi: 10.1142/S0218216511009212. [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. [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. [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. [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. [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. [13] D. Calegari, MSJ Memoirs, Vol. 20, Mathematical Society of Japan, Tokyo, 2009. doi: 10.1142/e018. [14] E. I. Dinaburg, A connection between various entropy characterizations of dynamical systems, Izv. Akad. Nauk SSSR Ser. Mat., 35 (1971), 324-366. [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. [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. [17] J.-M. Gambaudo and É. Ghys, Commutators and diffeomorphisms of surfaces, Ergodic Theory Dynam. Systems, 24 (2004), 1591-1617. doi: 10.1017/S0143385703000737. [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. [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. [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. [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. [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. [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. [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. [25] S. Schleimer, Notes on the complex of curves, http://homepages.warwick.ac.uk/ masgar/Maths/notes.pdf. [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. [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. [28] Y. Yomdin, Volume growth and entropy, Israel J. Math., 57 (1987), 285-300. doi: 10.1007/BF02766215. [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.
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